OBSTRUCTIONS FOR EMBEDDING GRAPHS INTO

SURFACES

by

Petr Skoda

M.Sc., Charles University, 2009

B.Sc., Charles University, 2007

a Thesis submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

in the

Department of Mathematics

Faculty of Science

c© Petr Skoda 2012

SIMON FRASER UNIVERSITY

Fall 2012

This work is licensed under the Creative Commons

Attribution-NonCommercial-NoDerivs 2.5 Canada

(http://creativecommons.org/licenses/by-nc-nd/2.5/ca/)

APPROVAL

Name: Petr Skoda

Degree: Doctor of Philosophy

Title of Thesis: Obstructions for Embedding Graphs into Surfaces

Examining Committee: Malgorzata Dubiel, Senior Lecturer

Chair

Bojan Mohar, Professor

Senior Supervisor

Matt DeVos, Assistant Professor

Supervisor

Luis Goddyn, Professor

Supervisor

Ladislav Stacho, Associate Professor

SFU Examiner

Frederic Havet

INRIA Sophia-Antipolis

External Examiner

Date Approved: September 13th, 2012

ii

Partial Copyright Licence

iii

Abstract

Only for two surfaces, the 2-sphere and the projective plane, the complete list of obstructions

is known. We aim to expand our understanding of obstructions for higher-genus surfaces

by studying obstructions of low connectivity. Classes of graphs are described such that

each obstruction of connectivity 2 is obtained as a 2-sum of graphs from those classes.

In particular, this structure allows us to determine the complete lists of obstructions of

connectivity 2 for the torus and the Klein bottle.

iv

To my parents for letting me fly.

v

Acknowledgments

Thank you, Bojan, for teaching me the academic skills and sharing with me your vast

knowledge. There are so many things I learned on top of a mountain.

vi

Contents

Approval ii

Partial Copyright License iii

Abstract iv

Dedication v

Acknowledgments vi

Contents vii

List of Tables ix

List of Figures x

1 Introduction 1

1.1 Bridges and cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Surfaces and embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 2-Sums and graphs with terminals . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Graph parameters and critical graphs . . . . . . . . . . . . . . . . . . . . . . 7

2 Torus 9

2.1 Alternating genus and genus of 2-sums . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Critical classes for graph parameters . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Hoppers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Dumbbells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

vii

2.5 General orientable surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6 Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.7 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Graphs Critical for the Alternating Genus 39

3.1 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Basic classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3 XY-labelled graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 Connectivity 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5 Connectivity 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.6 The class C0(ga) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4 The Klein Bottle 64

4.1 Euler genus of 2-sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 Critical classes, cascades, and hoppers . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Euler genus 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.4 The Klein bottle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5 Cascades 83

5.1 Separating cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2 Disjoint K-graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3 The class S1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4 Nonplanar extensions of planar bases . . . . . . . . . . . . . . . . . . . . . . . 100

6 Label Transitions Around a Vertex 109

6.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.2 Bounded Number of Labels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.3 Dealing with Bridge Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.4 NP-Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Bibliography 127

Indices 129

Index of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Index of Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

viii

List of Tables

2.1 Possible outcomes for a minor-operation in a g-tight part of a 2-sum. . . . . . 16

2.2 Possible outcomes for a minor-operation in a g+-tight part of a 2-sum. . . . . 18

2.3 Classification of g-tight parts of a 2-sum. . . . . . . . . . . . . . . . . . . . . . 29

2.4 Classification of g+-tight parts of a 2-sum. . . . . . . . . . . . . . . . . . . . . 31

2.5 Classification of parts of obstructions of connectivity 2 for the torus. . . . . . 36

4.1 Possible outcomes for a minor-operation in a g-tight part of a 2-sum. . . . . . 68

4.2 Classification of g-tight parts of a 2-sum. . . . . . . . . . . . . . . . . . . . . . 70

4.3 Classification of g-tight parts of a 2-sum in C◦2(g). . . . . . . . . . . . . . . . . 74

ix

List of Figures

2.1 Kuratowski graphs and their two-vertex alternating embeddings in the torus. 10

2.2 (a) An illustration of an embedding of the xy-sum of two xy-alternating

graphs on the torus. For better clarity, the vertices x and y were split into 5

vertices each. Contract all the edges incident with x and y to get the xy-sum.

(b) The 2-sum of two copies of K5 embedded into the torus. . . . . . . . . . . 12

2.3 Hasse diagram showing relations of several graph parameters. An edge indi-

cates that the values of parameters differ by at most one and the parameter

below is bounded from above by the parameter above. . . . . . . . . . . . . . 15

2.4 A sketch of the structure of the graph G from the proof of Lemma 2.28. . . . 28

2.5 The class C◦0(g+), the third graph is the sole member of the class C◦0(g). . . . . 32

3.1 A case in the proof of Lemma 3.6. . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 T5, splits of Kuratowski graphs which belong to C◦0(g+a) \ C◦0(ga) . . . . . . . . 45

3.3 An example of an XY-labelled graph and its corresponding graph in G◦xy. . . 45

3.4 T2, the xy-sums of graphs in C◦0(g+) which belong to C◦0(ga) ∩ C◦0(g+a). . . . . . 47

3.5 The XY-labelled representation of C◦0(g+). . . . . . . . . . . . . . . . . . . . . 48

3.6 The XY-labelled representation of T6 ⊆ C◦0(g+a) \ C◦0(ga). . . . . . . . . . . . . 49

3.7 The XY-labelled representation of T4 = C◦0(ga)\C◦0(g+a). The underlined labels

are used in the proof of Lemma 3.23. . . . . . . . . . . . . . . . . . . . . . . . 50

3.8 The XY-labelled representation of T3 ⊆ C◦0(ga)∩C◦0(g+a). For each white vertex

v ∈ V (G), we have g(G− v) = 1. . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.9 Venn diagram of critical classes (for alternating genus) for the torus. . . . . . 60

3.10 The graph Pinch minus a vertex. The white vertices form one part of the

K3,3-subdivision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

x

3.11 An embedding of G+ in the torus for a graph G corresponding to a 3-

alternating graph in LXY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1 The class C◦0(g+), the third graph is the sole member of the class C◦0(g). . . . . 72

5.1 The planar graphs in C◦1(g+). . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2 Linkages to small sets. In each linkage, any subset of feet can be contracted. . 91

5.3 (a) A graph with Lx induced by x, v0, v1, v2 and Ly induced by y, v3, v4, v5, v6, v9.

The set U = {v1, v2, v9} separates Ly from Lx but not Lx from Ly. (b)

A U -linkage with core Ly and feet v4v10v2, v9, v6v8v1. Each of the feet

v4v10v2 and v9 is removable but v6v8v1 is not. This shows that Ly and U

admit linkage (5.2g) (shown in Fig. 5.2(g)) using F = {v3v9, v1v8, v2v10} and

u1 7→ v1, u2 7→ v9, u3 7→ v2. (c) A completion of the linkage. . . . . . . . . . . 92

5.4 Selected nonplanar graphs in C◦1(g+). . . . . . . . . . . . . . . . . . . . . . . . 96

5.5 Cascades in S1 whose xy-K-graphs are 1-separated. . . . . . . . . . . . . . . . 97

5.6 Cascades in S1 whose xy-K-graphs are 2-separated. . . . . . . . . . . . . . . . 98

5.7 Set B of bases of cascades in S1 whose xy-K-graphs are k-separated for k ≥ 3. 99

5.8 The class B∗ \ B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.9 Cascades in S1 whose xy-K-graphs are k-separated for k ≥ 3. . . . . . . . . . 108

xi

Chapter 1

Introduction

The problem which graphs can be embedded in a given surface is a fundamental question

in topological graph theory. Robertson and Seymour [23] proved that for each surface S the

class of graphs that embed into S can be characterized by a finite list Forb(S) of minimal

forbidden minors (or obstructions). For the 2-sphere S0, Forb(S0) consists of the Kuratowski

graphs, K5 and K3,3. The list of obstructions Forb(N1) for the projective plane N1 already

contains 35 graphs and N1 is the only other surface for which the complete list of obstructions

is known. The number of obstructions for both orientable and nonorientable surfaces seems

to grow fast with the genus and that can be one of the reasons why even for the torus S1

the complete list of obstructions is still not known, although thousands of obstructions were

generated by computer (see [13]).

It is easy to show that obstructions for orientable surfaces that are not 2-connected

can be obtained as disjoint unions and 1-sums of obstructions for surfaces of smaller genus

(see Theorem 2.1). For nonorientable surfaces the structure of obstructions that are not

2-connected is not so simple but still straightforward (see [27]). In this thesis, we study the

obstructions for surfaces that are 2-connected but have a 2-vertex-cut. In Chapter 2, a list

of obstructions of connectivity 2 for the torus is constructed and shown to be complete. In

Chapter 4, a list of obstructions of connectivity 2 for the Klein bottle is obtained and shown

to be complete.

1

CHAPTER 1. INTRODUCTION 2

1.1 Bridges and cycles

In this thesis, we will deal mainly with the class G of simple graphs. Let G ∈ G be a simple

graph. A branch vertex of G is a vertex of G of degree different from 2. A branch P is a path

connecting two branch vertices v1, v2 such that all vertices in V (P ) \ {v1, v2} have degree 2.

An open branch is obtained from a branch by removing its endvertices. A subdivision of G

is a graph obtained from G by replacing each edge of G by a path of length at least 1. A

graph H is homeomorphic to G if H is isomorphic to a subdivision of G.

Let H be a subgraph of G. An H-bridge B in G is a subgraph of G that is either induced

by an edge in E(G) \ E(H) with both ends in H or consists of a connected component C

of G − V (H) together with all edges (and their vertices) which have one end in C. In the

former case we say that B is trivial . The vertices in V (B) ∩ V (H) are the attachments of

B. We also say that B attaches at v, for v ∈ V (B)∩ V (H). The subgraph B◦ = B − V (H)

of G is the interior of B. Thus, G − B◦ is the graph obtained from G by removing the

H-bridge B. The bridge B is a local bridge if all attachments of B lie on a single branch of

H.

Let C be a cycle of a fixed orientation and u and v two vertices in C. The segment

C[u, v] is the path P in C from u to v (in the given orientation of C). Similarly, C(u, v)

denotes P without the endvertices and any combination of brackets can be used to indicate

which endvertices are included in the path. Let P be a segment of C and B a C-bridge

whose attachments are contained in P . The support of B in P is the smallest subsegment

of P that contains all attachments of B.

Let C be a cycle in a graph G. Two C-bridges B1 and B2 overlap if at least one of the

following conditions hold:

(i) B1 and B2 have three attachments in common.

(ii) C contains distinct vertices v1, v2, v3, v4 that appear in this order on C such that v1

and v3 are attachments of B1 and v2 and v4 are attachments of B2.

In the case (ii), we say that B1 and B2 skew-overlap. If B1, B2 do not overlap, they avoid

each other and there are vertices u, v ∈ V (C) such that all attachments of B1 lie on C[u, v]

and all attachments of B2 lie on C[v, u]. A C-bridge B is planar if C ∪B is planar. Let Bbe the set of C-bridges. The overlap graph O(G,C) has vertex set B and two C-bridges are

adjacent if they overlap. We use the following well-known theorem (see e.g. [4]).

CHAPTER 1. INTRODUCTION 3

Theorem 1.1. Let G be a graph and C a cycle in G such that all C-bridges are planar.

Then G is planar if and only if O(G,C) is bipartite.

1.2 Surfaces and embeddings

A surface S is a two-dimensional topological manifold (see [20] for a proper definition).

Surfaces come in two types: orientable and nonorientable. Let Sk be the orientable surface

obtained from the 2-sphere by adding k handles and let Nk be the nonorientable surface

obtained from the 2-sphere by adding k cross-caps, k ≥ 1. A fundamental result states

that each surface is homeomorphic either to Sk or Nk for some k ≥ 0.

Let G be a connected multigraph. An embedding of G into S is a mapping ψ that

assigns to each vertex of G a distinct point in S and each edge uv ∈ E(G) a simple arc with

ends ψ(u) and ψ(v) whose interior is disjoint from the images of other edges and vertices.

An embedding of G is 2-cell if each connected region of S − ψ(G) is homeomorphic to an

open disk. 2-cell embeddings can be described combinatorially as follows: A combinatorial

embedding Π of G is a mapping that assigns, to each vertex v ∈ V (G), a cyclic permutation

Π(v), called local rotation at v, of edges incident with v and assigns, to each edge e ∈ E(G),

a signature Π(e) ∈ {−1, 1}. Given a combinatorial embedding Π of G, we say that G is

Π-embedded .

Let G be a Π-embedded graph. A dart is a triple (u, uv, s) where uv ∈ E(G) and

s ∈ {−1, 1}. The traversal permutation τΠ is defined on the sets of darts of G as follows:

τΠ(u, uv, s) = (v,Π(v)sΠ(uv)(uv), sΠ(uv)).

If (u1, u1u2, s1), . . . , (uk, uku1, sk) is an orbit of τΠ, we say that the cyclic sequence u1, u1u2,

u2, . . . , uk, uku1 of vertices and edges is a Π-facial walk (or a Π-face).

Every edge appears either once in exactly two Π-facial walks or exactly twice in a single

Π-facial walk. The edges that appear twice in a Π-facial walk are called singular . Similarly,

if a vertex v appears more than once in a Π-facial walk W , then v is singular in W . If W

contains no singular vertices or edges, then W is a cycle and we call it a Π-facial cycle. Let

e1, e2 ∈ E(G) be two edges incident with u ∈ V (G). If e1, u, e2 appears as a subsequence in

a Π-facial walk W , we call the pair e1, e2 a Π-angle of W at u.

Let G be a Π-embedded graph. A cycle C of G is Π-one-sided if it contains an odd

number of edges with negative signature. Otherwise C is Π-two-sided . If G contains a

CHAPTER 1. INTRODUCTION 4

Π-one-sided cycle, then the embedding Π is nonorientable. Otherwise Π is orientable.

Let G be a Π-embedded graph and let F (Π) be the set of Π-facial walks. The Euler

genus g(Π) of Π is defined by Euler’s formula as

|V (G)| − |E(G)|+ |F (Π)| = 2− g(Π). (1.1)

The Euler genus g(G) of G is the minimum Euler genus of an embedding of G. The

(orientable) genus g(G) is half of the minimum Euler genus of an orientable embedding of

G. If G contains at least one cycle, then the nonorientable genus g(G) is the minimum

Euler genus of a nonorientable embedding of G, else g(G) = 0. The following relation is

an easy observation (see [20]).

Lemma 1.2. For every connected graph G which is not a tree,

g(G) ≤ 2g(G) + 1.

If g(G) = 2g(G) + 1, then G is said to be orientably simple. Note that in this case

g(G) = 2g(G) = g(G)− 1, i.e., the Euler genus of G is even.

Let G ∈ G be a simple graph and e an edge of G. Then G−e denotes the graph obtained

from G by deleting e and G/e denotes the graph1 obtained from G by contracting e. It is

convenient for us to formalize these graph operations. The setM(G) = E(G)×{−, /} is the

set of minor-operations available for G. An element µ ∈M(G) is called a minor-operation

and µG denotes the graph obtained from G by applying µ. For example, if µ = (e,−) then

µG = G − e. A graph H is a minor of G if H can be obtained from a subgraph of G by

contracting some edges. If G is connected, then H can be obtained from G by a sequence

of minor-operations.

A graph G ∈ G such that G does not embed into S but each proper minor of G does is

called an obstruction for S. A graph G ∈ G of minimal degree 3 such that G does not embed

into S but each subgraph of G does is called a topological obstruction for S. Let Forb(S) be

the set of obstructions for S and let Forb∗(S) be the set of topological obstructions for S.

A classic result of Robertson and Seymour [23] asserts that both Forb(S) and Forb∗(S) are

finite for each surface S.

For a fixed Kuratowski graph K, a Kuratowski subgraph in G is a minimal subgraph of

G that contains K as a minor. A K-graph in G is a subgraph L of G which is homeomorphic

1When contracting an edge, one may obtain multiple edges. We shall replace any multiple edges by singleedges as such a simplification has no effect on the genus.

CHAPTER 1. INTRODUCTION 5

to either K4 or K2,3 and there is an L-bridge in G that attaches to all four branch vertices

of L when L is homeomorphic to K4 or attaches to all three open branches of L when L is

homeomorphic to K2,3. Such an L-bridge is a dfprincipal L-bridge. We are using extensively

the following well-known theorem which states that Forb∗(S0) = {K5,K3,3}. Wagner [29]

showed that Forb(S0) = Forb∗(S0).

Theorem 1.3 (Kuratowski [19]). A graph is planar if and only if it does not contain a

subdivision of a Kuratowski graph as a subgraph.

Let H0 be a subdivision of K3,3, let v be a branch vertex of H0, and let u1, u2, u3 be the

neighbors of v. The graph H = H0 − v is called a tripod . The three (possibly trivial) paths

in H with ends u1, u2, u3, respectively, are the feet of H. We say that H is attached to a

subgraph K of G if H is contained in a K-bridge B, u1, u2, u3 are attachments of B, and B

has no other attachments.

Let C be a subgraph of a graph G. A path in G whose endpoints belong to C but is

internally disjoint from C is called a C-path. Let C be a cycle and let P1 and P2 be two

C-paths with ends u1, v1 and u2, v2, respectively. If u1, u2, v1, v2 are distinct vertices of C

and appear on C in this (interlaced) order, then we say that P1 and P2 are crossing C-paths.

We use the following classic theorem (see [20, Theorem 6.3.1]).

Theorem 1.4. Let G be a connected graph and C a cycle in G. Let G′ be a graph obtained

from G by adding a new vertex joined to all vertices of C. Then G can be embedded in plane

with C as an outer cycle unless G contains an obstruction of the following type:

(i) a pair of disjoint crossing C-paths,

(ii) a tripod attached to C, or

(iii) a Kuratowski subgraph contained in a 3-connected block of G′ distinct from the 3-

connected block of G′ containing C.

The following observation is useful.

Lemma 1.5. Let uvw be a triangle in a graph G. If u has degree 3 in G, then every

embedding of G− vw into a surface can be extended into an embedding of G into the same

surface.

CHAPTER 1. INTRODUCTION 6

Proof. Let Π be an embedding of G − vw into a surface S. Since u has degree 3, v and w

share a Π-face W . The embedding obtained from Π by embedding the edge vw inside W is

an embedding of G into S.

1.3 2-Sums and graphs with terminals

A graph G is k-connected if G has at least k + 1 vertices and G remains connected after

deletion of any k−1 vertices. A graph has connectivity k if it is k-connected but not (k+1)-

connected. An edge whose deletion increases the number of connected components of the

graph is a cutedge. A cutvertex of G is a vertex whose removal increases the number of

components of G. A vertex is nonseparating if it is not a cutvertex.

In this thesis, we study obstructions for embedding graphs into surfaces that have con-

nectivity 2. Given graphs G1 and G2 such that V (G1) ∩ V (G2) = {x, y}, we say that the

graph G = (V (G1) ∪ V (G2), E(G1) ∪ E(G2)) is the xy-sum of G1 and G2. The graphs G1

and G2 are the parts of the xy-sum. If x and y are not important, we sometimes refer to G

as a 2-sum.

We wish to study the parts of a 2-sum separately and, in order to do so, we mark the

vertices of the separation as terminals. This prompts us to study the class Gxy of graphs

with two terminals, x and y. The letters x and y will be consistently used for the two

distinguished terminals. Most notions that are used for graphs can be used in the same way

for graphs with terminals. Some notions differ though and, to distinguish between graphs

with and without terminals, let G be the underlying graph of G without terminals (for

G ∈ Gxy). Two graphs, G1 and G2, in Gxy are isomorphic, also denoted G1∼= G2, if there

is an isomorphism of the graphs G1 and G2 that maps terminals of G1 onto terminals of

G2 (and non-terminals onto non-terminals) possibly exchanging x and y. We define minor-

operations on graphs in Gxy in the way that Gxy is a minor-closed class. When performing

edge contractions on G ∈ Gxy, we do not allow contraction of the edge xy (if xy ∈ E(G)) and

when contracting an edge incident with a terminal, the resulting vertex becomes a terminal.

We use M(G) to denote the set of available minor-operations for G. Since (xy, /) 6∈ M(G)

for G ∈ Gxy, we shall use G/xy to denote the underlying simple graph in G obtained from G

by identification of x and y; for this operation, we do not require the edge xy to be present

in G. Let vxy be the vertex obtained after the identification. For convenience, we use G◦xyfor the subclass of Gxy of graphs without the edge xy. We use G+ for the graph G plus the

CHAPTER 1. INTRODUCTION 7

edge xy if it is not already present.

We will use the following lemma (see [20, Prop. 6.1.2.]).

Lemma 1.6. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy. If G+2 is planar, then

each embedding of G+1 into a surface can be extended to an embedding of G+ into the same

surface.

Proof. Since G+2 is planar, there is a planar embedding of G2 such that x and y are on the

infinite face. A given embedding Π of G+1 can be extended into the embedding of G+ by

embedding G2 into a Π-face incident with the edge xy.

1.4 Graph parameters and critical graphs

A graph parameter is a function G → R that is constant on each isomorphism class of

G. Similarly, we call a function Gxy → R a graph parameter if it is constant on each

isomorphism class of Gxy. A graph parameter P is minor-monotone if P(H) ≤ P(G) for

each graph G ∈ Gxy and each minor H of G. All the genera defined above (Euler, orientable,

nonorientable) are minor-monotone graph parameters. Several other graph parameters will

be used in this thesis.

Let P be a graph parameter. A graph G is P-critical if P(µG) < P(G) for each

µ ∈ M(G). Let H be a subgraph of a graph G (possibly with terminals) and P a graph

parameter. We say thatH is P-tight if P(µG) < P(G) for every minor-operation µ ∈M(H).

We observe that every subgraph of a P-critical graph is P-tight:

Lemma 1.7. Let H1, . . . ,Hs be subgraphs of a graph G (possibly with terminals). If E(H1)∪· · · ∪ E(Hs) = E(G), then G is P-critical if and only if H1, . . . ,Hs are P-tight.

For a graph parameter P, let C(P) denote the class of graphs G ∈ Gxy that are P-critical.

We call C(P) the critical class for P. Let C◦(P) be the subclass of C(P) of graphs without

the edge xy. We refine the class C(P) according to the value of P. Let Ck(P) denote the

subclass of C(P) that contains precisely the graphs G for which P(G) = k + 1. The classes

C◦k(P) are defined similarly as subclasses of C◦(P). A P-critical graph has threshold k if

P(G) > k and P(µG) ≤ k for each µ ∈M(G).

For a graph parameter P, we say that a minor-operation µ ∈ M(G) decreases P by at

least k if P(µG) ≤ P(G)−k. The subset ofM(G) that decreases P by at least k is denoted

CHAPTER 1. INTRODUCTION 8

by ∆k(P, G). We write just ∆k(P) when the graph is clear from the context. Note that G

is P-critical if and only if M(G) = ∆1(P).

Two graph parameters P and Q are r-separated (in this order) if P(G) ≤ Q(G) ≤P(G) + r for all graphs G ∈ Gxy. If L = Q − P, then we also say that P and Q are

r-separated by L. It is easy to see that, for k ≥ 0 and G ∈ Gxy, if P and Q are r-separated

by L and L(G) = d, then the following holds:

(S1) ∆k+r−d(P) ⊆ ∆k(Q) and

(S2) ∆k+d(Q) ⊆ ∆k(P).

Chapter 2

Torus

The structure of obstructions for orientable surfaces that have connectivity at most 1 is very

simple. They are disjoint unions and 1-sums of obstructions for surfaces of smaller genus.

This can be easily seen as an application of the following theorem that states that the genus

of graphs is additive with respect to their 2-connected components (or blocks).

Theorem 2.1 (Battle et al. [3]). The genus of a graph is the sum of the genera of its blocks.

Stahl [26] and Decker et al. [11] showed that genus of 2-sums differs by at most 1 from

the sum of genera of its parts. Decker et al. [12] provided a simple formula introduced

in Section 2.1. In this chapter, we shall prove that each obstruction of connectivity 2 for

an orientable surface can be obtained as a 2-sum of building blocks that fall (roughly)

into two classes of graphs. One class consists of obstructions for embedding into surfaces

of smaller genus. The graphs in the second class, C(ga), are critical with respect to the

graph parameter ga defined in Section 2.1. We use this characterization in Section 2.6 to

construct all obstructions of connectivity 2 for the torus. The class C0(ga) which appears in

the construction is determined in Chapter 3. We conclude the chapter with a list of open

problems in Section 2.7. In this chapter, all embeddings and genera are orientable.

2.1 Alternating genus and genus of 2-sums

Let G ∈ Gxy. The genus of G+ can be also viewed as a graph parameter g+ defined as

g+(G) = g(G+). The graph parameter θ = g+−g captures the difference between the genera

of G+ and G, that is θ(G) = g+(G)− g(G). Note that θ(G) ∈ {0, 1}.

9

CHAPTER 2. TORUS 10

x

y

x

y1

y2

Figure 2.1: Kuratowski graphs and their two-vertex alternating embeddings in the torus.

In order to compute the genus of an xy-sum, it is necessary to know whether G has a

minimum genus embedding Π with x and y appearing at least twice in an alternating order

in a Π-face. More precisely, we say that an embedding Π is xy-alternating if there is a Π-face

W such that (x, y, x, y) is a cyclic subsequence of W . A graph G ∈ Gxy is xy-alternating if it

admits a minimum genus embedding that is xy-alternating. Fig. 2.1 shows two examples of

xy-alternating embeddings in the torus. We associate a graph parameter with this property.

Let ε(G) = 1 if G is xy-alternating and ε(G) = 0 otherwise. We shall also use the graph

parameter ε+ defined as ε+(G) = ε(G+).

In order to describe minimum genus embeddings of the xy-sum G of graphs G1, G2 ∈ Gxy,it is sufficient to consider two types of embeddings. To construct them, we take particular

minimum genus embeddings Π1 and Π2 of G1 and G2 (respectively) and combine them into

an embedding Π of G. For a non-terminal vertex v, let the local rotation around v in Π

be the same as the local rotation around v in Πi (if v ∈ V (Gi) for i ∈ {1, 2}). Consider

Π1-faces W1 and W2 incident with x and y, respectively, and Π2-faces W3 and W4 incident

with x and y, respectively. Note that the faces W1 and W2 (and also W3 and W4) need not

to be distinct. We distinguish three cases.

Case 1: W1,W2,W3,W4 are distinct faces.

Write the face W1 as (x, e1, U1, e2), W2 as (y, f1, U2, f2), W3 as (x, e3, U3, e4), and W4

as (y, f3, U4, f4), where U1, . . . , U4 are subwalks of W1, . . . ,W4, respectively. Let e1, S1, e2

be the linear sequence obtained from Π1(x) by cutting it at e1, e2 (where S1 is a linear

subsequence of Π1(x)). Similarly, let e3, S2, e4 be the linear sequence obtained from Π2(x)

by cutting it at an e3, e4. We let Π(x) be the cyclic sequence (e1, S1, e2, e3, S2, e4). Similarly,

we define Π(y) as the concatenation of the two linear sequences obtained from Π1(y) and

CHAPTER 2. TORUS 11

Π2(y) by cutting each of them at f1, f2 and f3, f4, respectively. Each Π1-face and Π2-

face different from W1,W2,W3, and W4 is also a Π-face. The faces W1 and W3 combine

into the Π-face (x, e1, U1, e2, x, e3, U3, e4) and the faces W2 and W4 combine into the Π-face

(y, f1, U2, f2, y, f3, U4, f4). Thus, the total number of faces decreased by two and (1.1) gives

the following value h0(G) of g(Π):

g(Π) = g(G1) + g(G2) + 1 =: h0(G). (2.1)

Case 2: W1,W2,W3,W4 consist of three distinct faces.

We may assume that W3 = W4 = (x, e3, U3, f4, y, f3, U4, e4). The same construction as

in the previous case (with W1 and W2 expressed as above) combines W1,W2, and W3 into

a single Π-face (x, e1, U1, e2, e3, U3, f4, y, f1, U2, f2, y, f3, U4, e4, x). Again, the total number

of faces decreases by two and the genus of Π is given by (2.1).

Case 3: W1 = W2 and W3 = W4.

Observe that since W1 = W2, we have that θ(G1) = 0 and, similarly, we have θ(G2) = 0.

Write W1 = W2 = (x, e1, U1, f2, y, f1, U2, e2) and W3 = W4 = (x, e3, U3, f4, y, f3, U4, e4).

The above construction combines W1 and W3 into the Π-faces (x, e1, U1, f2, y, f3, U4, e4)

and (y, f1, U2, e2, x, e3, U3, f4). Thus, the total number of faces did not change and (1.1)

gives the following value of g(Π).

g(Π) = g(G1) + g(G2). (2.2)

Suppose that Π1 and Π2 are minimum genus embeddings of G1 and G2 (respectively)

that are both xy-alternating. Let W1 and W2 be the xy-alternating faces of Π1 and

Π2, respectively, and write W1 as (x, e1, U1, f2, y, f1, U2, e4, x, e3, U3, f4, y, f3, U4, e2) and W2

as (x, e5, U5, f6, y, f5, U6, e8, x, e7, U7, f8, y, f7, U8, e6). Again, the local rotation Π(v) of a

non-terminal vertex v ∈ V (Gi) is set to Πi(v), i = 1, 2. To construct Π(x), cut Π1(x)

at e1, e2 and e3, e4 to obtain two linear sequences e1, S1, e4 and e3, S2, e2 and cut Π2(x)

at e5, e6 and e7, e8 to obtain e5, S3, e8 and e7, S4, e6. Let Π(x) be the cyclic sequence

(e1, S1, e4, e5, S3, e8, e3, S2, e2, e7, S4, e6). We construct Π(y) similarly. Fig. 2.2 illustrates

this process and gives an example of a 2-sum of two K5’s. The faces W1 and W2 are combined

into Π-faces (x, e1, U1, f2, y, f7, U8, e6), (y, f1, U2, e4, x, e5, U5, f6), (x, e3, U3, f4, y, f5, U6, e8),

CHAPTER 2. TORUS 12

x

y

e1

f2

f5

e8

e4

f1

f8

e7

e3

f4

f7

e6

e2

f3

f6

e5

S1

S2

S3S4y

x

Figure 2.2: (a) An illustration of an embedding of the xy-sum of two xy-alternating graphson the torus. For better clarity, the vertices x and y were split into 5 vertices each. Contractall the edges incident with x and y to get the xy-sum. (b) The 2-sum of two copies of K5

embedded into the torus.

and (y, f3, U4, e2, x, e7, U7, f8). As the total number of faces increased by two, (1.1) gives

the following value of g(Π).

g(Π) = g(G1) + g(G2)− 1. (2.3)

Usually, there is a minimum genus embedding of G constructed from the minimum genus

embeddings of G1 and G2. Suppose now that θ(G1) = 1, ε+(G1) = 1, and ε(G2) = 1. Since

θ(G1) = 1, the only embedding described above that we can construct from minimum genus

embeddings of G1 and G2 has genus g(G1)+g(G2)+1. On the other hand, g(G+1) = g(G1)+1

and both G+1 and G2 are xy-alternating. Thus we obtain an embedding of G of genus

g(G+1) + g(G2)− 1 = g(G1) + g(G2) < g(G1) + g(G2) + 1. Hence it is necessary to consider

also the embeddings of G+1 and G+

2 . The minimum of the genera given by equations (2.2)

and (2.3) can be combined into a single value, denoted h1(G):

h1(G) = g+(G1) + g+(G2)− ε+(G1)ε+(G2). (2.4)

Using the parameters defined above, we can write

h1(G) = g(G1) + g(G2) + θ(G1) + θ(G2)− ε+(G1)ε+(G2).

The similarity of the above equation to (2.1) leads us to define the graph parameter

η(G1, G2) = θ(G1) + θ(G2) − ε+(G1)ε+(G2). Note that η(G1, G2) ∈ {−1, 0, 1, 2}. This

CHAPTER 2. TORUS 13

gives another expression for h1:

h1(G) = g(G1) + g(G2) + η(G1, G2). (2.5)

Decker et al. [12] proved the following formula for the genus of a 2-sum of graphs.

Theorem 2.2 (Decker, Glover, and Huneke [12]). Let G be the xy-sum of connected graphs

G1, G2 ∈ Gxy. Then,

(i) g(G) = min{h0(G), h1(G)},

(ii) g+(G) = h1(G),

(iii) ε+(G) = 1 if and only if ε+(G1) 6= ε+(G2), and

(iv) θ(G) = 1 if and only η(G1, G2) = 2.

Often, we consider minor-operations in the graph G1 while the graph G2 is fixed. When

ε+(G2) = 1, the genus of G depends on the graph parameter ga = g−ε, called the alternating

genus of G. Let g+a = g+− ε+ be the graph parameter defined as g+

a(G) = ga(G+) =

g+(G) − ε+(G). If we know the value of the parameter ε+(G2), then we can express h1(G)

as follows. If ε+(G2) = 1, then (2.4) can be rewritten as

h1(G) = g+a(G1) + g+(G2). (2.6)

Else, (2.4) is equivalent to

h1(G) = g+(G1) + g+(G2). (2.7)

The next lemma shows that alternating genus is a minor-monotone graph parameter.

Lemma 2.3. Let G ∈ Gxy. If H is a minor of G, then ga(H) ≤ ga(G).

Proof. If g(H) < g(G) or ε(H) ≥ ε(G), then the result trivially holds. Hence if the claimed

inequality is violated, then g(H) = g(G), ε(H) = 0, and ε(G) = 1. Thus, there is an

xy-alternating minimum genus embedding Π of G. Let W0 be an xy-alternating Π-face.

We may assume without loss of generality that H is obtained from G by a single minor-

operation. Suppose first that H = G− e for some edge e ∈ E(G). Let Π′ be the embedding

of H induced by Π. If e is a singular edge that appears in a Π-face W , then W is split

into two Π′-faces in Π′. Thus g(H) ≤ g(Π′) = g(Π) − 1 = g(G) − 1 which contradicts the

CHAPTER 2. TORUS 14

assumption that g(H) = g(G). Hence e appears in two different Π-faces W1 and W2. The

faces W1 and W2 combine to form a single Π′-face W ′ in Π′. Thus g(Π′) = g(Π). As either

W0 is a Π′-face or W0−e is a subsequence of W ′, we conclude that Π′ is also xy-alternating.

This contradicts the assumption that g(H) = g(G) and ε(H) = 0.

Suppose now that H = G/e for some edge e ∈ E(G). Let Π′ be the induced embedding

of H obtained from Π by contracting e. That is, the local rotation Π′(ve) around the vertex

ve obtained by contraction of e = uv is set to be the concatenation of the linear sequences

obtained from Π(u) and Π(v) by cutting them at e. If e does not appear in W0, then W0

is also a Π′-face. Otherwise, as e 6= xy, Π′ contains a facial walk W ′0 that can be obtained

from W0 by replacing each (of at most 2) occurrence of u, e, v by ve. It is immediate that

W ′0 is an xy-alternating Π′-face. This again contradicts the choice of H.

The following lemma shows how the property of being xy-alternating can be expressed

in terms of θ(G) and ε(G+).

Lemma 2.4. Let G ∈ G◦xy. The graph G is xy-alternating if and only if θ(G) = 0 and G+

is xy-alternating. In symbols, ε(G) = 1 if and only if θ(G) = 0 and ε+(G) = 1.

Proof. Assume that G is xy-alternating and let Π be an xy-alternating embedding of G

of genus g(G). By embedding the edge xy into the xy-alternating Π-face, we obtain an

embedding of G+ into the same surface that is also xy-alternating. This shows that θ(G) = 0

and ε+(G) = 1.

For the converse, assume that θ(G) = 0 and that G+ is xy-alternating. Let Π be an

xy-alternating embedding of G+ with an xy-alternating Π-face W . Since θ(G) = 0, the

edge xy is not a singular edge. Thus by deleting xy from Π, we obtain an embedding Π′

of G in the same surface where either W is a Π′-face or W − xy is a subsequence of a

Π′-face. Hence Π′ is an xy-alternating embedding of G. Since g(Π′) = g(G), the graph G is

xy-alternating.

Fig. 2.3 shows the relationship between the parameters g, g+, ga, and g+a . In addition to

the constraints given in the figure, there is one more interrelationship that is described by

the following lemma.

Lemma 2.5. For a graph G ∈ Gxy, we have either ga(G) = g(G) or ga(G) = g+a(G).

CHAPTER 2. TORUS 15

ǫ+θ

ǫ

g+a

g+

g

ga

Figure 2.3: Hasse diagram showing relations of several graph parameters. An edge indicatesthat the values of parameters differ by at most one and the parameter below is boundedfrom above by the parameter above.

Proof. If ε(G) = 0, then ga(G) = g(G) and we are done. Otherwise, ε(G) = 1 and Lemma 2.4

gives that ε+(G) = 1 and θ(G) = 0. Therefore, g+a(G) = ga(G) + ε(G) + θ(G) − ε+(G) =

ga(G).

We shall show in this section that each minor-operation in a 2-connected g-tight part of

an xy-sum decreases at least one of the graph parameters g, g+, and g+a . Note that several

parameters can be decreased by a single minor-operation and it depends on the relations

between the parameters. For example, if G is K3,3 with the terminals that are non-adjacent

and we consider an edge e of G, then the contraction (e, /) belongs both to ∆1(g) and ∆1(g+)

as g(G/e) = g+(G/e) = 0. But G is xy-alternating (see Fig. 2.1), so ga(G) = g+a(G) = 0 and

(e, /) belongs neither to ∆1(ga) nor to ∆1(g+a).

Several graph parameters defined above are 1-separated (see Fig. 2.3). The parameters

g and g+ are 1-separated by the parameter θ, ga and g by ε, g+a and g+ by ε+, and ga and g+

a

are 1-separated by the parameter ε + θ − ε+. We shall prove formally only that ga and g+

are 1-separated by ε+ θ. As g+ = ga + ε+ θ it is enough to show the following.

Lemma 2.6. For any graph G ∈ Gxy, we have that

g+(G) ≤ ga(G) + 1.

Proof. By Lemma 2.5, either ga(G) = g(G) or ga(G) = g+a(G). In the former case, g+(G) =

g(G) + θ(G) ≤ ga(G) + 1. In the latter case, g+(G) = g+a(G) + ε+(G) ≤ ga(G) + 1.

Let P and Q be graph parameters 1-separated by L and k ≥ 0. For 1-separated param-

eters, the statements (S1) and (S2), on page 7, can be rewritten as follows:

CHAPTER 2. TORUS 16

ε+(G2) η(G1, G2) µ

00 ∆1(g+)1 ∆1(g) or ∆1(g+)2 ∆1(g)

1

-1 ∆1(g+a)

0 ∆2(g) or ∆1(g+a)

1 ∆1(g) or ∆1(g+a)

2 ∆1(g) or ∆2(g+a)

Table 2.1: Possible outcomes for a minor-operation in a g-tight part of a 2-sum.

(S1) ∆k+1(P) ⊆ ∆k(Q) and, if L(G) = 1, then ∆k(P) ⊆ ∆k(Q);

(S2) ∆k+1(Q) ⊆ ∆k(P) and, if L(G) = 0, then ∆k(Q) ⊆ ∆k(P).

Using the new notation we can state the following corollary of Lemma 2.5.

Corollary 2.7. For each G ∈ Gxy, ∆1(ga) ⊆ ∆1(g) ∪∆1(g+a).

Proof. Let µ ∈ ∆1(ga). If µ 6∈ ∆1(g)∪∆1(g+a), then g(µG) = g(G) > ga(µG) and g+

a(µG) =

g+a(G) > ga(µG), which contradicts Lemma 2.5 (for the graph µG).

The next lemma describes necessary and sufficient conditions for a single part of a 2-sum

of graphs to be g-tight. This is a key lemma and its outcome, summarized in Table 2.1, will

be used heavily throughout this chapter.

Lemma 2.8. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy and µ ∈ M(G1) such

that µG1 is connected. Then g(µG) < g(G) if and only if the following is true (where ∆k(·)always refers to the decrease of the parameter in G1):

(i) If η(G1, G2) = −1, then µ ∈ ∆1(g+a).

(ii) If η(G1, G2) = 0, then µ ∈ ∆1(g+) when ε+(G2) = 0 and µ ∈ ∆2(g) ∪ ∆1(g+a) when

ε+(G2) = 1.

(iii) If η(G1, G2) = 1, then µ ∈ ∆1(g) ∪∆1(g+) when ε+(G2) = 0 and µ ∈ ∆1(g) ∪∆1(g+a)

when ε+(G2) = 1.

(iv) If η(G1, G2) = 2, then µ ∈ ∆1(g) when ε+(G2) = 0 and µ ∈ ∆1(g) ∪ ∆2(g+a) when

ε+(G2) = 1.

CHAPTER 2. TORUS 17

Proof. Let us start with the “only if” part. Since µG1 is connected, Theorem 2.2 can be

used to determine g(µG). We will do the cases (i), (ii) and (iii) together. Assume that

η(G1, G2) ≤ 1. If ε+(G2) = 0, let us assume that µ 6∈ ∆1(g+) and if ε+(G2) = 1, let

us assume that µ 6∈ ∆1(g+a). By (2.6) and (2.7), h1(µG) = h1(G). Since g(µG) < g(G),

Theorem 2.2 gives that g(µG) = h0(G). By using the definition of h0(G) in (2.1), we obtain:

g(µG1) + g(G2) + 1 = h0(G) = g(µG) < g(G) = g(G1) + g(G2) + η(G1, G2).

Thus g(µG1) ≤ g(G1) + η(G1, G2) − 2. If η(G1, G2) = −1, then µ ∈ ∆3(g) which implies

that µ ∈ ∆2(g+) by (S1) as g and g+ are 1-separated. By (S2) applied to g+a and g+, we

obtain that µ ∈ ∆1(g+a), a contradiction. Thus (i) holds. If η(G1, G2) = 0, then µ ∈ ∆2(g).

This proves (ii) when ε+(G2) = 1. If ε+(G2) = 0, then µ ∈ ∆1(g+) by (S1). This yields (ii).

If η(G1, G2) = 1, then µ ∈ ∆1(g) which yields (iii).

Suppose now that η(G1, G2) = 2 and that µ 6∈ ∆1(g). Then h0(G) = h0(µG). By

Theorem 2.2 and (2.5), g(G) = h0(G). Since g(µG) < g(G), we conclude that g(µG) =

h1(µG). As η(G1, G2) = 2, we know that θ(G1) = θ(G2) = 1 and ε+(G1)ε+(G2) = 0. Thus

we may write

g(G) = h0(G) = g(G1) + g(G2) + 1 = g+(G1) + g+(G2)− 1.

If ε+(G2) = 0, then we obtain using (2.7) that

g+(µG1) + g+(G2) = g(µG) < g(G) = g+(G1) + g+(G2)− 1.

Hence µ ∈ ∆2(g+) which implies by (S2) that also µ ∈ ∆1(g), a contradiction.

If ε+(G2) = 1, then ε+(G1) = 0 and g+a(G1) = g+(G1). We use (2.6) to obtain that

g+a(µG1) + g+(G2) = g(µG) < g(G) = g+

a(G1) + g+(G2)− 1.

Hence µ ∈ ∆2(g+a) and (iv) holds. This finishes the “only if” part.

To prove the “if” part, we assume that (i)–(iv) hold and show that g(µG) < g(G). We

start by proving that, if µ ∈ ∆1(g) and η(G1, G2) ≥ 1, then g(µG) < g(G). By Theorem 2.2,

g(G) = h0(G). Since g(µG) ≤ h0(µG), we obtain that

g(µG) ≤ g(µG1) + g(G2) + 1 < g(G1) + g(G2) + 1 = g(G).

If µ ∈ ∆2(g) and η(G2, G2) = 0, we have a similar inequality:

g(µG) ≤ g(µG1) + g(G2) + 1 < g(G1) + g(G2) = g(G).

CHAPTER 2. TORUS 18

ε+(G2) µ

0 ∆1(g+)1 ∆1(g+

a)

Table 2.2: Possible outcomes for a minor-operation in a g+-tight part of a 2-sum.

Similarly, we do the cases when µ ∈ ∆1(g+) and when µ ∈ ∆1(g+a). Suppose that

µ ∈ ∆1(g+), ε+(G2) = 0, and η(G1, G2) ≤ 1. By Theorem 2.2, g(G) = h1(G). We obtain

from Theorem 2.2 and (2.7) that

g(µG) ≤ g+(µG1) + g+(G2) < g+(G1) + g+(G2) = g(G).

Suppose now that µ ∈ ∆1(g+a), ε+(G2) = 1, and η(G1, G2) ≤ 1. We obtain from Theo-

rem 2.2 and (2.6) that

g(µG) ≤ g+a(µG1) + g+(G2) < g+

a(G1) + g+(G2) = g(G).

In the remaining case, when η(G2, G2) = 2, ε+(G2) = 1, and µ ∈ ∆2(g+a), we have a

similar inequality:

g(µG) ≤ g+a(µG1) + g+(G2) < g+

a(G1) + g+(G2)− 1 ≤ g(G1) + g(G2) + 1 = g(G).

This finishes the proof of the lemma.

We have a similar lemma for g+-tight part of a 2-sum, summarized in Table 2.2.

Lemma 2.9. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy and µ ∈ M(G1) such

that µG1 is connected. If ε+(G2) = 0, then g+(µG) < g+(G) if and only if µ ∈ ∆1(g+). If

ε+(G2) = 1, then g+(µG) < g+(G) if and only if µ ∈ ∆1(g+a).

Proof. Since µG is connected, Theorem 2.2 asserts that g+(G) and g+(µG) are equal to

h1(G) and h1(µG), respectively. Assume first that g+(µG) < g+(G). If ε+(G2) = 0, then

by (2.7),

g+(µG1) + g+(G2) = h1(µG) = g+(µG) < g+(G) = h1(G) = g+(G1) + g+(G2).

Thus g+(µG1) < g+(G1), yielding that µ ∈ ∆1(g+). If ε+(G2) = 1, then by (2.6),

g+a(µG1) + g+(G2) = h1(µG) = g+(µG) < g+(G) = h1(G) = g+

a(G1) + g+(G2).

CHAPTER 2. TORUS 19

Hence g+a(µG1) < g+

a(G1), yielding that µ ∈ ∆1(g+a).

Assume now that µ ∈ ∆1(g+) and ε+(G2) = 0. We obtain from Theorem 2.2 and (2.7)

that

g+(µG) = g+(µG1) + g+(G2) < g+(G1) + g+(G2) = g+(G).

Assume now that µ ∈ ∆1(g+a) and ε+(G2) = 1. We obtain from Theorem 2.2 and (2.6) that

g+(µG) = g+a(µG1) + g+(G2) < g+

a(G1) + g+(G2) = g+(G).

This completes the proof of the lemma.

Since for each graph precisely one hypothesis in the cases (i)–(iv) of Lemma 2.8 holds,

we obtain the following corollary.

Corollary 2.10. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy and µ ∈ M(G1)

such that µG1 is connected and g(µG) < g(G). Then µ ∈ ∆1(g) ∪∆1(g+) ∪∆1(g+a). Fur-

thermore, if ε+(G2) = 0, then µ ∈ ∆1(g) ∪∆1(g+).

Lemmas 2.8 and 2.9 characterize a graph with two terminals that is part of an obstruction

for an orientable surface. The next lemma describes when the edge xy is g-tight in G+ for

an xy-sum G.

Lemma 2.11. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy and H = G+. Then

the subgraph of H induced by the edge xy is g-tight if and only if η(G1, G2) = 2 and either

g(G1/xy) < g+(G1) or g(G2/xy) < g+(G2).

Proof. By Theorem 2.2(iv), θ(G) = 1 if and only if η(G1, G2) = 2. Thus g(H − xy) < g(H)

if and only if η(G1, G2) = 2. We may thus assume that η(G1, G2) = 2.

Theorem 2.1 implies that

g(H/xy) = g(G1/xy) + g(G2/xy).

Since ε+(G1)ε+(G2) = 0, Theorem 2.2 and (2.4) gives that

g(H) = g+(G) = h1(G) = g+(G1) + g+(G2).

Therefore, g(H/xy) < g(H) if and only if g(G1/xy) + g(G2/xy) < g+(G1) + g+(G2). Since

g(G1/xy) ≤ g+(G1) and g(G2/xy) ≤ g+(G2), we obtain that g(H/xy) < g(H) if and only if

g(G1/xy) < g+(G1) or g(G2/xy) < g+(G2).

CHAPTER 2. TORUS 20

2.2 Critical classes for graph parameters

Lemmas 2.8 and 2.9 provide necessary and sufficient conditions on the parts of an xy-sum

for being g-tight and g+-tight. In this section, we shall study and categorize some of the

graphs that satisfy these conditions, the graphs that belong to the classes C◦(g), C◦(g+),

C◦(ga), and C◦(g+a).

It is easy to see that, for each graph G ∈ C◦k(g), the graph G is an obstruction for Sk. On

the other hand, for each graph G ∈ Forb(Sk) and two non-adjacent vertices x and y of G,

the graph in Gxy obtained from G by making x and y terminals belongs to C◦k(g). Similarly

to C◦k(g), the class C◦k(g+) can be constructed from the graphs in Forb(Sk).

Lemma 2.12. Let G ∈ C◦k(g+). If θ(G) = 0, then G ∈ Forb(Sk). If θ(G) = 1, then either

G+∈ Forb(Sk), or G+∈ Forb∗(Sk) and G/xy ∈ Forb(Sk).

Proof. If θ(G) = 0, then M(G) = ∆1(g) by (S2) and thus G ∈ C◦k(g). Therefore G ∈Forb(Sk) as explained above. Suppose now that θ(G) = 1. Since G ∈ C◦(g+), we have that

M(G) ⊆ ∆1(g,G+). As g(G+− xy) < g(G+) and all other minor-operations in G+ except

contracting the edge xy decrease the genus of G+), there are no vertices of degree 2 in G and

deletion of each edge in G+ decreases g(G+). Thus G+ ∈ Forb∗(Sk). If g(G/xy) < g+(G),

then G+ ∈ Forb(Sk) since both deletion and contraction of xy decreases the genus of G+.

On the other hand, if g(G/xy) = g+(G), take any minor-operation µ ∈ M(G/xy). Since µ

is also a minor-operation in G, we obtain that g(µ(G/xy)) ≤ g+(µG) < g+(G) = g(G/xy)

as µ(G/xy) is a minor of µG+. Since µ was chosen arbitrarily, G/xy ∈ Forb(Sk).

Since the parameters g and g+ are 1-separated, the graphs whose minor-operations de-

crease either g or g+ belong either to C◦(g) or to C◦(g+).

Lemma 2.13. Let G ∈ G◦xy. If M(G) = ∆1(g) ∪∆1(g+), then G belongs to either C◦(g) or

C◦(g+).

Proof. If θ(G) = 0, then ∆1(g+) ⊆ ∆1(g) by (S2). Thus M(G) = ∆1(g) and G ∈ C◦(g).

Similarly, if θ(G) = 1, then ∆1(g) ⊆ ∆1(g+) by (S1). We conclude that M(G) = ∆1(g+)

and G ∈ C◦(g+).

The classes C◦(ga) and C◦(g+a) are closely related to the class C(ga) as we shall see below.

Let G∗ denote the graph obtained from G ∈ Gxy as the xy-sum of G and K5−xy (Fig. 2.5a).

CHAPTER 2. TORUS 21

The next lemma shows that the graphs in Ck(ga) are derived from graphs in Forb∗(Sk) and

thus the finiteness of Forb∗(Sk) implies the finiteness of Ck(ga).

Lemma 2.14. Let G ∈ Ck(ga). Then precisely one of the graphs G, G+, or G∗ belongs to

Forb∗(Sk).

Proof. For each e ∈ E(G) (possibly e = xy), we have that g(G−e) ≤ k and, if g(G−e) = k,

then G− e is xy-alternating on Sk. If g(G) > k, then G ∈ Forb∗(Sk), since g(G− e) ≤ k for

each e ∈ E(G). Thus we may assume that g(G) = k and G is not xy-alternating on Sk.Suppose that g(G+) > g(G). For each e ∈ E(G), we have either g(G − e) = k − 1 and

thus g(G+− e) = k, or G− e is xy-alternating on Sk and then g(G+− e) = k since the edge

xy can be embedded into the xy-alternating face. Therefore, G+∈ Forb∗(Sk).Suppose now that g(G+) = g(G). We shall show that G∗ ∈ Forb∗(Sk). Since G is not

xy-alternating on Sk, g(G∗) > g(G) by Lemma 3.1. For e ∈ E(G), either g(G− e) = k − 1

and thus g(G∗−e) ≤ k by Theorem 2.2, or G−e is xy-alternating on Sk and so g(G∗−e) = k

also by Theorem 2.2. Let H be the xy-bridge of G∗ isomorphic to (Fig. 2.5a). For e ∈ E(H),

since H+− e is planar, Lemma 1.6 gives that g(G∗ − e) = g(G+) = g(G) = k. This shows

that G∗ ∈ Forb∗(Sk).In conclusion, G, G+, or G∗ belongs to Forb∗(Sk), and it is clear that only one of these

graphs is in Forb∗(Sk) since G ≤ G+ and G∗ has a subgraph homeomorphic to G+.

We have the following corollary of Lemma 2.14.

Corollary 2.15. For k ≥ 1, the class of graphs Ck(ga) is finite.

Proof. Let F be the class of all graphs with two terminals obtained from graphs H ∈Forb∗(Sk) by declaring two vertices of H to be terminals (in all possible ways), by declaring

two adjacent vertices to be terminals and deleting the edge joining them, or by removing a

bridge isomorphic to K5 minus an edge and declaring its two vertices of attachments to be

terminals. By Lemma 2.14, Ck(ga) ⊆ F . Since Forb∗(Sk) is finite, we have that F is finite

and thus Ck(ga) is finite.

Corollary 2.15 can also be proved directly using the following deep result. The classic

result of Robertson and Seymour [23] asserts that graphs are well-quasi-ordered under the

minor relation, that is, in each infinite set of graphs, there are two that are minor of each

other. An application of this result (private communication with Matt DeVos) gives that

CHAPTER 2. TORUS 22

Ck(ga) is finite for each k. This can be seen as follows: Assume that there exists k such

that the class Ck(ga) is infinite. Since, for each G ∈ Ck(ga), g(G) ≤ ga(G) = k + 1, none

of the graphs in Ck(ga) contains a complete graph Kc of size c = 8 + k as a minor. Let

C be the subclass of G obtained from Ck(ga) by attaching a copy of Kc to each of the two

terminals of a graph G ∈ Ck(ga) and “forgetting” the terminals. The result of Robertson

and Seymour implies that there are two graphs G1, G2 ∈ C such that G1 is a minor of G2.

The only Kc-minors in G2 are the two cliques of size c. Thus to obtain a subgraph of G2

isomorphic to G1 we cannot delete or contract the edges of the copies of Kc. It is not hard

to see that this is only possible if the original graphs in Ck(ga) are minors (as graphs in Gxy)of each other.

The classes C◦(ga) and C◦(g+a) are related to the class C(ga). The following lemma implies

that both C◦k(ga) and C◦k(g+a) are finite.

Lemma 2.16. Let G ∈ G◦xy and k ≥ 0. Then the following holds:

(i) G ∈ Ck(ga) if and only if G ∈ C◦k(ga).

(ii) G+∈ Ck(ga) if and only if G ∈ C◦k(g+a) \ C◦k(ga).

Proof. The statement (i) is an easy observation. Suppose that G+∈ Ck(ga). It is immediate

that G ∈ C◦k(g+a). Since ga(G) = ga(G

+−xy) < ga(G+) = k+1, the graph G does not belong

to C◦k(ga).

Suppose now that G ∈ C◦k(g+a) \ C◦k(ga). By definition, M(G) = ∆1(ga, G

+). If ga(G) =

g+a(G), then M(G) = ∆1(ga) by (S2) and it follows that G ∈ C◦k(ga). Thus ga(G) < g+

a(G).

Hence ga(G+) > ga(G) = ga(G

+− xy) and (xy,−) ∈ ∆1(ga, G+). We conclude that G+ ∈

Ck(ga) as ga(G+) = g+

a(G) = k + 1.

Also the graphs that do not belong to C◦(g+a) can be characterized.

Lemma 2.17. If G ∈ C◦(ga), then G 6∈ C◦(g+a) if and only if there exists µ ∈ M(G) such

that µ ∈ ∆1(g) \∆1(g+a).

Proof. The “if” part follows from the fact thatM(G) 6= ∆1(g+a). The “only if” part follows

from Corollary 2.7 as there is µ ∈M(G) such that µ 6∈ ∆1(g+a).

Corollary 2.7 says that each minor-operation that decreases alternating genus also de-

creases g or g+a . We have the following weakly converse statement.

CHAPTER 2. TORUS 23

Lemma 2.18. Let G ∈ G◦xy. If M(G) = ∆1(g) ∪∆1(g+a), then G belongs to at least one of

C◦(g), C◦(ga), or C◦(g+a).

Proof. By Lemma 2.5, either ga(G) = g(G) or ga(G) = g+a(G). If g(G) = g+

a(G) = ga(G),

then ∆1(g) ⊆ ∆1(ga) and ∆1(g+a) ⊆ ∆1(ga) by (S2). Thus G ∈ C◦(ga).

If g(G) > ga(G), then ∆1(g+a) ⊆ ∆1(ga) by (S2). By (S1), ∆1(ga) ⊆ ∆1(g). We conclude

that G ∈ C◦(g). Similarly, if g+a(G) > ga(G), then ∆1(g) ⊆ ∆1(ga) by (S2). By (S1),

∆1(ga) ⊆ ∆1(g+a). We conclude that G ∈ C◦(g+

a).

2.3 Hoppers

In this section, we describe three subclasses of C◦(g+) all of which we call hoppers. If G is

a graph in C◦(g+a) such that ε+(G) = 1, then we call G a hopper of level 2 . It is immediate

by (S1) that G ∈ C◦(g+). The level should indicate the difficulty to construct such a graph.

Hoppers of level 0 and 1 appear as parts of obstructions of connectivity 2 and are defined

below.

A graph G ∈ G◦xy is a hopper of level 1 if M(G) = ∆1(g+a) ∪ ∆2(g) and G 6∈ C◦(g+

a).

Similarly, a graph G ∈ G◦xy is a hopper of level 0 ifM(G) = ∆1(g)∪∆2(g+a) and G 6∈ C◦(g).

Let Hl, 0 ≤ l ≤ 2, denote the class of hoppers of level l. Let Hlk denote the subclass of Hl

of graphs G with g+(G) = k.

Lemma 2.19. If G ∈ H0, then G ∈ C◦(g+), ε+(G) = 0, and θ(G) = 1.

Proof. By (S1), ∆2(g+a) ⊆ ∆1(g+). If θ(G) = 0, then ∆1(g+) ⊆ ∆1(g) by (S2) — a contra-

diction with G 6∈ C◦(g). Hence θ(G) = 1. By (S1), ∆1(g) ⊆ ∆1(g+) and we conclude that

G ∈ C◦(g+).

If ε+(G) = 1, then ∆2(g+a) ⊆ ∆2(g+) by (S1) and, since ∆2(g+) ⊆ ∆1(g) by (S2), we have

that ∆2(g+a) ⊆ ∆1(g), a contradiction. Thus ε+(G) = 0.

Note that the proof of the next lemma is analogous to the proof of Lemma 2.19.

Lemma 2.20. If G ∈ H1, then G ∈ C◦(g+), ε+(G) = 1, and θ(G) = 0.

Proof. By (S1), ∆2(g) ⊆ ∆1(g+). If ε+(G) = 0, then ∆1(g+) ⊆ ∆1(g+a) by (S2) — a

contradiction with G 6∈ C◦(g+a). Hence ε+(G) = 1. By (S1), ∆1(g+

a) ⊆ ∆1(g+) and we

conclude that G ∈ C◦(g+).

CHAPTER 2. TORUS 24

If θ(G) = 1, then ∆2(g) ⊆ ∆2(g+) by (S1) and, since ∆2(g+) ⊆ ∆1(g+a) by (S2), we have

that ∆2(g) ⊆ ∆1(g+a), a contradiction. Thus θ(G) = 0.

Similarly to orientable genus, alternating genus decreases by at most 1 when an edge is

deleted.

Lemma 2.21. Let G ∈ Gxy. For each e ∈ E(G), ga(G− e) ≥ ga(G)− 1.

Proof. Suppose that ga(G − e) < ga(G) − 1. Since g(G − e) ≥ g(G) − 1, we have that

ε(G) = 0, ε(G− e) = 1, and g(G− e) = g(G)− 1. Let Π be an xy-alternating embedding of

G− e in Sk, k = g(G− e) and let W be an xy-alternating Π-face. If the endvertices u and

v of e are Π-cofacial, then Π can be extended to an embedding of G in Sk, a contradiction.

Otherwise, let Π′ be the embedding of G on Sk+1 obtained from Π by embedding e onto a

new handle connecting faces incident with u and v. Since W is a subwalk of a Π′-face, Π′

is xy-alternating. Since g(Π′) = g(G − e) + 1 = g(G), we have that ε(G) = 1 which is a

contradiction.

Lemma 2.21 has the following corollary that shows the motivation for introducing the

notion of hoppers of level 2.

Corollary 2.22. A graph G ∈ Ck(ga) does not embed into Sk+1 if and only if ε+(G) = 1.

We conjecture that all graphs in Ck(ga) embed into Sk+1, that is, there are no hoppers

of level 2.

Conjecture 2.23. Each G ∈ Ck(ga) embeds into Sk+1.

Lemma 3.23 shows that Conjecture 2.23 is true for k = 0. The following lemma shows

that Conjecture 2.23 is true if xy ∈ E(G).

Lemma 2.24. Let G ∈ G◦xy. Then ga(G) < g+a(G) if and only if ε+(G) = 0 and θ(G) = 1.

Proof. If ga(G) < g+a(G), then ga(G) = g(G) by Lemma 2.5. Since ga and g+

a are 1-separated,

g+a(G) − ga(G) = 1 = ε(G) + θ(G) − ε+(G). Since ε(G) = 0, we obtain that θ(G) = 1 and

ε+(G) = 0 as required.

If ε+(G) = 0 and θ(G) = 1, then ε(G) = 0 by Lemma 2.4. Thus ga(G) < ga(G) + ε(G) +

θ(G)− ε+(G) = g+a(G).

Lemmas 2.16 and 2.24 assert that H2k ⊆ C◦k(ga).

CHAPTER 2. TORUS 25

2.4 Dumbbells

Lemmas 2.8 and 2.9 provide information about the minor-operations in a part of an xy-sum

except for a deletion of an edge that disconnects the graph. Since the xy-sum is 2-connected,

deletion of such a cutedge of G1 separates x and y. In this section, we determine how g-tight

and g+-tight parts of an xy-sum with such a cutedge look like.

If G1 ∈ G◦xy and b ∈ E(G1) is a cutedge of G1 whose deletion separates x and y, we say

that G1 is a dumbbell with bar b.

Lemma 2.25. If G1 is a dumbbell with bar b, then ε+(G1) = 0, (b, /) 6∈ ∆1(g) ∪ ∆1(g+),

and (b,−) 6∈ ∆1(g) ∪∆1(ga).

Proof. Since g(G1 − b) = g(G1) and ε(G1 − b) = 0 (as the terminals of G1 − b are not con-

nected), (b,−) 6∈ ∆1(g) ∪∆1(ga). Suppose for a contradiction that ε+(G1) = 1; then there

exists an xy-alternating minimum-genus embedding Π of G+1 . Let W be an xy-alternating Π-

facial walk. The walkW can be split into four subwalks containing x and y. Each of the edges

xy and b appears precisely twice in the Π-facial walks (either once in two different Π-facial

walks or twice in a single Π-facial walk). Since each walk from x to y has to use either xy or b,

both xy and b are singular edges that appear twice inW . Since Π is an orientable embedding,

the edge xy appears inW once in the direction from x to y and once from y to x. Hence, there

is another appearance of one of the terminals, say x, in W that is not incident with the edge

xy. We can write W as W = (x, xy, y,W1, e1, x, e2,W2, y, xy, x, e3,W3, e4). The local rota-

tion around x can be written as (xy, e4, S1, e2, e1, S2, e3). Let Π′ be the embedding obtained

from Π by letting Π′(v) = Π(v) for v ∈ V (G1) \ {x} and Π(x) = (e4, S1, e2, xy, e1, S2, e3).

All Π-facial walks except W are also Π′-facial walks as all Π-angles not incident with W

are also Π′-angles. The Π-facial walk W is split into three Π′-facial walks: (x, xy, y,W1, e1),

(x, e3,W3, e4), and (x, e2,W2, y, xy). Thus g(Π′) < g(Π), a contradiction with Π being a

minimum-genus embedding of G+1 . We conclude that ε+(G1) = 0.

Let µ = (b, /) be the contraction operation of b in G1. We shall show that µ 6∈ ∆1(g) ∪∆1(g+). Let H1 and H2 be the components of G1 − b. By Theorem 2.1, g(G1) = g(H1) +

g(H2) = g(µG1). If b is incident with a terminal, say b = zy, z ∈ V (H1), then G+1 is the

1-sum of H1 + b+ xy and H2. By Theorem 2.1,

g(G+1) = g(H1 + b+ xy) + g(H2) = g(H1 + xz) + g(H2) = g(µG+

1).

Thus g+(G1) = g+(µG1).

CHAPTER 2. TORUS 26

Suppose that b is not incident with a terminal and let z ∈ V (H1) be an endpoint of b.

Consider the graphs H ′1 = H1 + xy and H ′2 = H2 + b as members of the class G◦yz. Observe

that H ′1 and H ′2 are dumbbells (in G◦yz). We have already shown that ε+(H ′1) = ε+(H ′2) = 0

and g(µH ′2) = g(H ′2) and, since the bar of H ′2 is incident with a terminal, g+(µH ′2) = g+(H ′2).

By Theorem 2.2 (when G+1 is viewed as a yz-sum of H ′1 and H ′2),

g(G+1) = min{g(H ′1) + g(H ′2) + 1, g+(H ′1) + g+(H ′2)}, and

g(µG+1) = min{g(H ′1) + g(µH ′2) + 1, g+(H ′1) + g+(µH ′2)}.

Since g(µH ′2) = g(H ′2) and g+(µH ′2) = g+(H ′2), we conclude that g(µG+1) = g(G+

1). Thus

g+(µG1) = g+(G1). This shows that µ 6∈ ∆1(g) ∪∆1(g+).

Lemma 2.26. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy. If G1 is a dumbbell

with bar b and G1 is g-tight or g+-tight in G, then ε+(G1/b) = 1 and b is unique, that is, G1

has a single cutedge separating x and y.

Proof. By Lemmas 2.9, 2.25, and Corollary 2.10, (b, /) ∈ ∆1(g+a) \∆1(g+). It is immediate

that ε+(G1/b) = 1.

For the second part, suppose that there is another bar e 6= b in G1. By Lemma 2.25,

ε+(G1/b) = 0 as G1/b is a dumbbell with bar e, a contradiction. We conclude that b is

unique.

Let D be the class of dumbbells G1 with bar b such that θ(G1) = 0, ε+(G1/b) = 1, and

M(G1) \ {(b,−), (b, /)} ⊆ ∆1(g).

Lemma 2.27. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy such that G1 is a

dumbbell. Then G1 is g-tight in G if and only if ε+(G2) = 1, η(G1, G2) = 1, and one of the

following holds:

(i) G1 ∈ C◦(g+a).

(ii) G1 ∈ D.

Furthermore, G1 is g+-tight in G if and only if ε+(G2) = 1 and G1 ∈ C◦(g+a).

Proof. By Lemmas 2.25 and 2.26, ε+(G1) = 0 and G1 has a unique bar b for which it holds

that (b, /) 6∈ ∆1(g) ∪∆1(g+) and ε+(G1/b) = 1. Hence g+a(G1/b) = g+

a(G1)− 1 and we have

that (b, /) ∈ ∆1(g+a) \∆2(g+

a).

CHAPTER 2. TORUS 27

Assume that G1 is g-tight in G. By Corollary 2.10, ε+(G2) = 1. Assume first that

θ(G1) = 1. We shall show that (i) holds. Since (b, /) 6∈ ∆1(g) ∪ ∆2(g+a), Lemma 2.8 gives

that η(G1, G2) ≤ 1. Since ε+(G1) = 0 and θ(G1) = 1, we conclude that η(G1, G2) = 1. It

remains to show that G1 ∈ C◦(g+a).

Since θ(G1) = 1, we have that g+(G1 − b) = g(G+1 − b) = g(G1) < g+(G1) and thus

(b,−) ∈ ∆1(g+). Since ε+(G1) = 0, (S2) gives that (b,−) ∈ ∆1(g+a).

Let µ ∈ M(G1) \ {(b,−), (b, /)}. Since µG1 is connected, Lemma 2.8 gives that µ ∈∆1(g) ∪∆1(g+

a). By (S1), ∆1(g) ⊆ ∆1(g+). By (S2), ∆1(g+) ⊆ ∆1(g+a). We conclude that

µ ∈ ∆1(g+a). Since µ was arbitrary and (b,−), (b, /) ∈ ∆1(g+

a), we have thatM(G1) = ∆1(g+a)

and G1 ∈ C◦(g+a). Therefore, (i) holds.

Assume now that θ(G1) = 0. We shall show that (ii) holds. In G−b, the two components

of G1 − b are joined to G2 by single vertices. If η(G2) = 0, Theorems 2.1 and 2.2 imply

(using ε+(G1) = 0 and θ(G1) = 0) that

g(G− b) = g(G1 − b) + g(G2) = g(G1) + g(G2) = g+(G1) + g+(G2) = h1(G) = g(G).

Thus η(G1, G2) ≥ 1. Since θ(G1) = 0, we conclude that θ(G2) = 1 and η(G1, G2) = 1.

It remains to show that G1 ∈ D, namely that µ ∈ ∆1(g) for each µ ∈ M(G) \{(b,−), (b, /)}. Let µ ∈M(G) \ {(b,−), (b, /)}. Since µG1 is connected, µ ∈ ∆1(g)∪∆1(g+

a)

by Lemma 2.8. Since µG is still a dumbbell, ε+(µG) = 0 by Lemma 2.25. Hence g+(µG) =

g+a(µG) and ∆1(g+

a) ⊆ ∆1(g+). By (S2), ∆1(g+) ⊆ ∆1(g). Therefore, µ ∈ ∆1(g). We

conclude that G1 ∈ D. Thus (ii) holds.

Assume now that ε+(G2) = 1 and η(G1, G2) = 1 and let us prove that if (i) or (ii) holds,

then G1 is g-tight. Assume first that (i) holds. Let µ ∈M(G1). We have that µ ∈ ∆1(g+a).

If µG1 is connected, g(µG) < g(G) by Lemma 2.8 since ε+(G2) = 1. Otherwise, µ = (b,−).

Since η(G1, G2) = 1, we obtain that

g(G− b) = g(G1 − b) + g(G2) = g(G1) + g(G2) < g(G1) + g(G2) + 1 = g(G).

Thus g(µG) < g(G) for each µ ∈M(G1). We conclude that G1 is g-tight in G.

Assume now that (ii) holds. Let µ ∈M(G1) and assume first that µG1 is connected. If

µ = (b, /), then (b, /) ∈ ∆1(g+a) since ε+(G1) = 0 and ε+(µG1) = 1. Otherwise µ ∈ ∆1(g)

since G1 ∈ D. Since η(G1, G2) = 1, and ε+(G2) = 1, Lemma 2.8 gives that g(µG) < g(G).

The case when µ = (b,−) remains. By Theorems 2.1 and 2.2,

g(G− b) = g(G1 − b) + g(G2) = g(G1) + g(G2) < g(G1) + g(G2) + 1 = g(G).

CHAPTER 2. TORUS 28

G1 G2

H1

H2

bz

x

y

Figure 2.4: A sketch of the structure of the graph G from the proof of Lemma 2.28.

We have that g(µG) < g(G) for each µ ∈M(G1). We conclude that G1 is g-tight.

Assume now that G1 is g+-tight. Since (b, /) 6∈ ∆1(g+), Lemma 2.9 gives that ε+(G2) = 1

and M(G) \ {b,−} ⊆ ∆1(g+a). By Theorems 2.1 and 2.2,

g(G1) + g+(G2) = g+(G− b) < g+(G) = g+(G1) + g+(G2).

Thus g(G1) < g+(G1) and

g+a(G1 − b(≤ g+(G1 − b) = g(G1) < g+(G1) = g+

a(G1).

We conclude that G1 ∈ C◦(g+a).

Assume now that ε+(G2) = 1 and G1 ∈ C◦(g+a). Let µ ∈ M(G) \ {(b,−)}. Since µG1 is

connected and µ ∈ ∆1(g+a), we have that g+(µG) < g+(G) by Lemma 2.9. Take µ = (b,−).

Since µ ∈ ∆1(g+a)\∆1(ga), we have that g+(G1) ≥ g+

a(G1) > ga(G1) = g(G1) by Lemma 2.5.

By Theorems 2.1 and 2.2,

g+(G− b) = g(G1 − b) + g(G+2) = g(G1) + g+(G2) < g+(G1) + g+(G2) = h1(G) = g+(G).

Hence G1 is g+-tight in G.

We close this section by showing that, in an obstruction of connectivity 2, there always

exists a 2-vertex-cut such that neither of the parts belongs to D.

Lemma 2.28. Let H ∈ Forb(Sk) be of connectivity 2. Then there exists a 2-vertex-cut

{x, y} ∈ V (H) and an xy-sum G such that H ∈ {G, G+} and neither of the parts of G

belongs to D.

Proof. Let G be an xy-sum of G1, G2 ∈ G◦xy such that H = {G, G+}. If H = G+, then G is

g+-tight and G1, G2 6∈ D by Lemma 2.27. We may thus assume that H = G. Suppose that

G1 ∈ D. Since G1 is g-tight in G and θ(G1) = 0, Lemma 2.27 gives that ε+(G2) = 1 and

η(G1, G2) = 1. From the definition of η(G1, G2) we conclude that θ(G2) = 1 as ε+(G1) = 0.

CHAPTER 2. TORUS 29

ε+(G2) η(G1, G2) G1 belongs to

00 C◦(g+)1 C◦(g) or C◦(g+)2 C◦(g)

1

-1 C◦(g+a)

0 C◦(g+a) or H1

1 C◦(g), C◦(ga), C◦(g+a), or D

2 C◦(g) or H0

Table 2.3: Classification of g-tight parts of a 2-sum.

Let b be a bar of G1 and let H1 and H2 be the components of G1− b. We may assume that

H1 contains at least one edge. Let x be the common vertex of H1 and G2 and let z be the

endpoint of b incident with H1. Let us view G as an xz-sum of H1 and G′2 = G2 ∪H2 + b

(see Fig. 2.4). We claim that neither H1 nor G′2 belongs to D.

By Lemma 2.26 applied to G1, b is the unique cutedge separating x and y and thus

there is no cutedge in H1 separating x and z. Therefore, H1 is not a dumbbell. We shall

show that θ(G′2) = 1 and hence G′2 6∈ D. The graph G′+2 can be viewed as the xy-sum of

G2 and the graph G′1 = H2 + b+ zx. The graph G′1 is a dumbbell and thus ε+(G′1) = 0 by

Lemma 2.25. By Theorem 2.1, g(G′2) = g(H2) + g(G2). By Theorem 2.2, using ε+(G1) = 0

and θ(G2) = 1,

g(G′+2 ) = min{g(G′1) + g(G2) + 1, g(G′+1 ) + g(G+2)} ≥ g(H2) + g(G2) + 1 = g(G′2) + 1.

Therefore θ(G′2) = 1. We conclude that G′2 6∈ D.

2.5 General orientable surface

In this section, we prove two general theorems that classify the g-tight and g+-tight parts

of an xy-sum. The classification of g-tight parts that is given in Theorem 2.29 below is also

summarized in Table 2.3.

Theorem 2.29. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy. The graph G1 is

g-tight if and only if the following statements hold:

(i) If η(G1, G2) = −1, then G1 ∈ C◦(g+a).

(ii) If η(G1, G2) = 0, then G1 ∈ C◦(g+) if ε+(G2) = 0 and G1 ∈ C◦(g+a) ∪H1 otherwise.

CHAPTER 2. TORUS 30

(iii) If η(G1, G2) = 1, then G1 ∈ C◦(g) ∪ C◦(g+) if ε+(G2) = 0 and G1 ∈ C◦(g) ∪ C◦(ga) ∪C◦(g+

a) ∪ D otherwise.

(iv) If η(G1, G2) = 2, then G1 ∈ C◦(g) if ε+(G2) = 0 and G1 ∈ C◦(g) ∪H0 otherwise.

Proof. Let us start with the “only if” part of the theorem. Assume first that G1 has no

cutedge that separates x and y. Lemma 2.8 classifies which graph parameters of G1 are

decreased by the minor-operations in M(G1). If it is a single parameter, G1 belongs to the

critical class corresponding to the parameter. For example, if ε+(G2) = 0 and η(G1, G2) = 0,

then M(G1) = ∆1(g+) by Lemma 2.8(i) and thus G1 ∈ C◦(g+). The statements (i), (ii) for

ε+(G2) = 0, and (iv) for ε+(G2) = 0 are proven in this way and we omit the details. Let us

focus on the remaining cases.

Let us start with the case when η(G1, G2) = 1. If ε+(G2) = 0, then M(G1) = ∆1(g) ∪∆1(g+) by Lemma 2.8(iii). By Lemma 2.13, G1 belongs to either C◦(g) or C◦(g+). If

ε+(G2) = 1, thenM(G1) = ∆1(g)∪∆1(g+a) by Lemma 2.8(iii). By Lemma 2.18, G1 belongs

to either C◦(g), C◦(ga), or C◦(g+a). This proves (iii).

If η(G1, G2) = 0 and ε+(G2) = 1, then M(G1) = ∆1(g+a) ∪ ∆2(g) by Lemma 2.8(ii).

By definition, G1 belongs to either C◦(g+a) or H1. Thus, (ii) holds. If η(G1, G2) = 2 and

ε+(G2) = 1, thenM(G1) = ∆1(g)∪∆2(g+a) by Lemma 2.8(iv). By definition, G1 belongs to

either C◦(g) or H0. Thus (iv) is true.

Assume now that G1 has a cutedge that separates x and y and thus G1 is a dumbbell.

Since G1 is g-tight, Lemma 2.27 gives that ε+(G2) = 1, η(G1, G2) = 1, and G1 ∈ C◦(g+a)∪D.

The statements (i), (ii), and (iv) are vacuously true since η(G1, G2) = 1. The statement

(iii) is true since G1 ∈ C◦(g+a) ∪ D. This completes the “only if” part of the proof.

It remains to prove the “if” part. Lemma 2.8 is now used to prove that G1 is g-

tight. Assume first that G1 has no cutedge separating x and y. If G1 belongs to one of

the classes C◦(g), C◦(g+), or C◦(g+a), then it is straightforward to check that in each case

Lemma 2.8 asserts that G1 is g-tight. We shall omit the proof here and do only the cases

when G1 ∈ C◦(ga) or G1 is a hopper.

If G1 ∈ C◦(ga), ε+(G2) = 1, and η(G1, G2) = 1, then Corollary 2.7 asserts thatM(G1) =

∆1(g) ∪ ∆1(g+a). Lemma 2.8 gives that G1 is g-tight. Finally, let us assume that G1 is

a hopper. If G1 ∈ H1, ε+(G2) = 1, and η(G1, G2) = 0, then M(G1) = ∆2(g) ∪ ∆1(g+a)

by definition of H1. Lemma 2.8 gives that G1 is g-tight. If G1 ∈ H0, ε+(G2) = 1, and

η(G1, G2) = 2, then M(G1) = ∆1(g) ∪∆2(g+a) by definition of H0. Lemma 2.8 gives that

CHAPTER 2. TORUS 31

ε+(G2) G1 belongs to

0 C◦(g+)1 C◦(g+

a)

Table 2.4: Classification of g+-tight parts of a 2-sum.

G1 is g-tight.

Assume now that G1 is a dumbbell with bar b. If G1 ∈ D (and ε+(G2) = 1, and

η(G1, G2) = 1), then G1 is g-tight by Lemma 2.27(ii). By Lemma 2.25, G1 6∈ C◦(g)∪C◦(g+)∪C◦(ga) and ε+(G1) = 0. Since H0 and H1 are subsets of C◦(g+), we have that G1 6∈ H0 ∪H1

(Lemmas 2.19 and 2.20). Thus we may assume that ε+(G2) = 1, G1 ∈ C◦(g+a) \ C◦(ga),

and (b,−) ∈ ∆1(g+a). Since ε+(G1 − b) = 0, (b,−) ∈ ∆1(g+). By (S2), θ(G1) = 1. Since

ε+(G1) = 0 and η(G1, G2) ≤ 1, we conclude that η(G1, G2) = 1. By Lemma 2.27(i), G1 is

g-tight in G. This completes the proof of the theorem.

Similarly, Theorem 2.30 classifies the g+-tight parts of an xy-sum and the outcome is

summarized in Table 2.4.

Theorem 2.30. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy. If ε+(G2) = 0,

then G1 is g+-tight if and only if G1 ∈ C◦(g+). If ε+(G2) = 1, then G1 is g+-tight if and only

if G1 ∈ C◦(g+a).

Proof. If G1 has no cutedge that separates x and y, then the result follows from Lemma 2.9.

Suppose now that G1 is a dumbbell. If ε+(G2) = 1, then G1 is g+-tight if and only if

G1 ∈ C◦(g+a) by Lemma 2.27. If ε+(G2) = 0, then G1 is not g+-tight by Lemma 2.27. By

Lemma 2.25, G1 6∈ C◦(g+). Thus the theorem holds for G1.

Note that a graph can belong to several critical classes at the same time. For example,

if G ∈ C◦(g) such that θ(G) = 1 and ε+(G) = 0, then G belongs to all four classes, C◦(g),

C◦(g+), C◦(ga), and C◦(g+a).

We finish this section by the following corollary which shows that at least one part of a

2-sum is an “obstruction” for a surface.

Corollary 2.31. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy. If both, G1 and

G2, are g-tight, then the following statements hold:

(i) G1 and G2 belong to C◦(g) ∪ C◦(g+) ∪ C◦(ga) ∪ C◦(g+a) ∪ D.

CHAPTER 2. TORUS 32

(a)

x y

(b)

x y

(c)

x y

Figure 2.5: The class C◦0(g+), the third graph is the sole member of the class C◦0(g).

(ii) If ε+(G2) = 0, then G1 ∈ C◦(g) ∪ C◦(g+).

(iii) Either G1 or G2 belongs to C◦(g) ∪ C◦(g+).

Proof. By Lemma 2.19 and 2.20, H0 andH1 are subclasses of C◦(g+). Thus (i) and (ii) follow

from Theorem 2.29 as it covers all possible combinations of the parameters describing G.

We shall now prove (iii). Assume that G2 does not belong to C◦(g) ∪ C◦(g+). If G2 is a

dumbbell, then Lemma 2.25 gives that ε+(G2) = 0 and thus G1 ∈ C◦(g) ∪ C◦(g+) by (ii).

Thus we may assume that µG2 is connected for each µ ∈ M(G2). Lemma 2.13 applied to

G2 gives that there exists a minor-operation µ ∈M(G2) such that µ 6∈ ∆1(g)∪∆1(g+). By

Corollary 2.10, µ ∈ ∆1(g+a). Since µ 6∈ ∆1(g+), we have that ε+(G2) = 0 by (S1). Therefore,

(ii) gives that G1 ∈ C◦(g) ∪ C◦(g+).

2.6 Torus

In this section, we characterize obstructions for embedding graphs of connectivity 2 into the

torus. We first show that the classes C◦0(g) and C◦0(g+) are related to Kuratowski graphs K5

and K3,3.

Lemma 2.32. The class C◦0(g) consists of a single graph, K3,3 with non-adjacent terminals

(Fig. 2.5c). The class C◦0(g+) consists of the three graphs shown in Fig. 2.5.

Proof. The obstructions Forb(S0) for the 2-sphere are K3,3 and K5. As we observed in

Section 2.2, a graph G belongs to C◦0(g) if only if G is isomorphic to a graph in Forb(S0)

with the terminals non-adjacent. Since xy 6∈ E(G), G cannot be isomorphic to K5, and

there is a unique 2-labeled graph isomorphic to K3,3 with two non-adjacent terminals.

Let us show first that each graph in Fig. 2.5 belongs to C◦0(g+). If G+ is isomorphic

to a Kuratowski graph, the lemma follows from the Kuratowski theorem. Otherwise G is

CHAPTER 2. TORUS 33

isomorphic to K3,3 with x and y non-adjacent. It suffices to show that µG+ is planar for

each minor-operation µ ∈ M(G) as G+ clearly embeds into the torus. Pick an arbitrary

edge e ∈ E(G). The graph G+− e has 9 edges and is not isomorphic to K3,3 as it contains a

triangle. The graph G+/e has only 5 vertices and (at most) 9 edges. Therefore both G+− eand G+/e contain no Kuratowski graph as a minor and are therefore planar. Since e was

arbitrary, it follows that µG+ is planar for every µ ∈M(G). We conclude that G ∈ C◦0(g+).

We shall show now that there are no other graphs in C◦0(g+). Let G ∈ C◦0(g+). By

Lemma 2.12, there is a graph H ∈ Forb∗(S0) such that either G is isomorphic to H or

G is isomorphic to the graph obtained from H by deleting an edge and making the ends

terminals. It is not hard to see that this yields precisely the graphs in Fig. 2.5.

Note that the first two graphs in Fig. 2.5 have θ equal to 1 and last one has θ equal to 0.

We summarize the properties of graphs in C◦0(g+) in the following lemma.

Lemma 2.33. For each graph G ∈ C◦0(g+), the graph G+ is xy-alternating on the torus,

G/xy is planar, and θ(G) = 1 if and only if G 6∈ C◦0(g).

Proof. By Lemma 2.32, G or G+ is isomorphic to a Kuratowski graph. The xy-alternating

embeddings of Kuratowski graphs are depicted in Fig. 2.1. Since each Kuratowski graph G

is xy-alternating for each pair of vertices of G, the graph G+ is also xy-alternating for each

pair of vertices of G by Lemma 2.4. For each Kuratowski graph G, the graph G/xy has at

most 5 vertices and at most 9 edges. Thus G/xy contains no Kuratowski graph as a minor

and is therefore planar.

Let us present some restrictions on 2-sums that are obstructions for the torus.

Lemma 2.34. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy and H ∈ {G, G+}. If

H ∈ Forb(S1), then

(i) g+(G1) = g+(G2) = 1,

(ii) ε+(G1)ε+(G2) = 0, and

(iii) η(G1, G2) = 2 if and only if H = G+.

Proof. Suppose that H ∈ Forb(S1). If g+(G1) ≥ 2, then G+1 contains a toroidal obstruction.

Since G+1 is a proper minor of H, this contradicts the fact that H ∈ Forb(S1). Thus

g+(G1) ≤ 1 and g+(G2) ≤ 1 by symmetry. If g+(G1) = 0, then g+(G) ≤ 1 by Lemma 1.6,

CHAPTER 2. TORUS 34

a contradiction. We conclude that g+(G1) = 1 and also g+(G2) = 1 by symmetry. This

shows (i).

If ε+(G1)ε+(G2) = 1, then it follows from Theorem 2.2 that

g(H) ≤ g+(G) = g+(G1) + g+(G2)− ε+(G1)ε+(G2) = 1,

a contradiction. Thus ε+(G1)ε+(G2) = 0 and (ii) holds.

To show (iii), suppose that H = G and η(G1, G2) = 2. By (i) and (ii), this is only

possible if g(G1) = g(G2) = 0. By Theorem 2.2, g(H) = g(G) ≤ g(G1) + g(G2) + 1 = 1, a

contradiction. The other implication follows from Lemma 2.11.

The classes C0(ga) and C0(g+a) are determined in Chapter 3. They are described using

six subclasses T1, . . . , T6 of G◦xy. Let T1 be the class of graphs that contains each G ∈ G◦xysuch that G is isomorphic to a Kuratowski graph plus one or two isolated vertices that are

terminals in G, T2 the class of graphs shown in Fig. 3.4, T3 the class of graphs corresponding

to the graphs in Fig. 3.8, T4 the class of graphs corresponding to the graphs in Fig. 3.7,

T5 the class of graphs depicted in Fig. 3.2, and T6 the class of graphs corresponding to the

graphs in Fig. 3.6. The graphs in T1 are disconnected and hence they do not appear in an

xy-sum of connectivity 2.

The following theorem is proven in Chapter 3.

Theorem 3.22. C◦0(ga) ∪ C◦0(g+a) = T1 ∪ · · · ∪ T6.

Furthermore,

(i) C◦0(ga) ∩ C◦0(g+a) = T1 ∪ T2 ∪ T3,

(ii) C◦0(ga) \ C◦0(g+a) = T4,

(iii) C◦0(g+a) \ C◦0(ga) = T5 ∪ T6,

Also, let us restate a corollary that is proven in Chapter 3 and which we shall use below.

Corollary 3.24. Let G be a graph in C◦0(ga) ∪ C◦0(g+a). Then g+(G) = 1 and ε+(G) = 0.

Moreover, θ(G) = 1 if and only if G ∈ C◦0(g+a) \ C◦0(ga).

It is time to present the main theorem of this section that derives a full characterization

of the obstructions of connectivity 2 for the torus. It can be viewed as an application of

Theorems 2.29 and 2.30 with the outcome summarized in Table 2.5.

CHAPTER 2. TORUS 35

Theorem 2.35. Suppose that G is the xy-sum of connected graphs G1, G2 ∈ G◦xy and that

the following statements hold:

(i) G1 ∈ C◦0(g+),

(ii) G2 ∈ C◦0(ga) ∪ C◦0(g+a),

(iii) if θ(G1) = θ(G2) = 0, then G2 ∈ C◦0(g+a).

If G1 ∈ C◦0(g) or G2 ∈ C◦0(ga), then G ∈ Forb(S1), else G+∈ Forb(S1). Furthermore, every

obstruction of connectivity 2 for the torus can be obtained this way.

Proof. The proof consists of two parts. In the first part, we prove that if G satisfies the

conditions (i)–(iii) then G or G+ is an obstruction for the torus. In the second part, all

obstructions of connectivity 2 are shown to be constructed this way.

Let us assume that (i)–(iii) holds. If G1 ∈ C◦0(g) or G2 ∈ C◦0(ga), let H = G, else let

H = G+. To show that H is an obstruction for the torus, Lemma 1.7 asserts that it is

sufficient to show that G1, G2, and xy (if xy ∈ E(H)) are g-tight in H and g(H) = 2. If

H = G, then this is equivalent to showing that G1, G2 are g-tight in G and g(G) = 2. If

H = G+, we need to show that that G1, G2 are g+-tight in G, g(G/xy) < g(G), θ(G) = 1,

and g+(G) = 2.

By (i) and Lemma 2.33, ε+(G1) = 1 and g+(G1) = 1. By (ii) and Corollary 3.24,

ε+(G2) = 0 and g+(G2) = 1. Hence h1(G) = 2. If η(G1, G2) = 2, then θ(G1) = θ(G2) = 1.

Thus G1 ∈ C◦0(g+) \ C◦0(g) by Lemma 2.33 and G2 ∈ C◦0(g+a) \ C◦0(ga) by Corollary 3.24.

Consequently, we have either η(G1, G2) ≤ 1 or H = G+. This excludes the case that

η(G1, G2) = 2 and H = G and we shall use it below. If H = G+, then by Theorem 2.2,

g(H) = g+(G) = h1(G) = 2 as required. Similarly, if H = G and η(G1, G2) ≤ 1, then

h1(G) ≤ h0(G) by (2.5). Hence g(H) = g(G) = h1(G) = 2 by Theorem 2.2.

It remains to prove the tightness. Since ε+(G2) = 0 and G1 ∈ C◦0(g+), Theorems 2.29

and 2.30 give that G1 is g-tight and g+-tight. If G2 ∈ C◦0(g+a), then G2 is g-tight and g+-tight

by Theorems 2.29 and 2.30 since ε+(G1) = 1. Otherwise, G2 ∈ C◦0(ga)\C◦0(g+a) and θ(G2) = 0

by Corollary 3.24. Thus H = G, θ(G1) = 1 by (iii), and we have that η(G1, G2) = 1. We

conclude that G2 is g-tight in H by Theorem 2.29.

If H = G, then we are done by Lemma 1.7. Suppose now that H = G+. It remains

to show that xy is g-tight in H. By assumption, we have that G1 ∈ C◦0(g+) \ C◦0(g) and

G2 ∈ C◦0(g+a)\C◦0(ga). Therefore, θ(G1) = 1 by Lemma 2.33 and θ(G2) = 1 by Corollary 3.24.

CHAPTER 2. TORUS 36

H ε+(G2) η(G1, G2) G1

G+0

—C◦0(g+)

1 C◦0(g+a)

G

00 C◦0(g+)1 C◦0(g) or C◦0(g+)

10 C◦0(g+

a)1 C◦0(ga) or C◦0(g+

a)

Table 2.5: Classification of parts of obstructions of connectivity 2 for the torus.

Hence η(G1, G2) = 2. Lemma 2.33 applied to G1 implies that g(G1/xy) < g+(G1). Thus

xy is g-tight in H by Lemma 2.11. We conclude that H is an obstruction for the torus by

Lemma 1.7.

Let us now prove that, for a graph H ∈ Forb(S1) of connectivity 2, there exists a 2-

vertex-cut {x, y} and an xy-sum G of graphs G1 and G2 such that H ∈ {G, G+} and the

statements (i)–(iii) hold. We pick x and y as guaranteed by Lemma 2.28 so that G1, G2 6∈ D.

If H = G, then G1 and G2 are g-tight in G. If H = G+, then G1 and G2 are g+-tight in

G and xy is g-tight in H. By Lemma 2.34, g+(G1) = g+(G2) = 1 and ε+(G1)ε+(G2) = 0.

We may assume by symmetry that ε+(G2) = 0. By Corollary 2.31(ii), the graph G1 belongs

to C◦0(g) ∪ C◦0(g+) = C◦0(g+) since g(G1) ≤ g+(G1) = 1. Hence (i) holds. By Lemma 2.33,

ε+(G1) = 1.

Since ε+(G2) = 0, Lemma 2.33 gives that G2 6∈ C◦0(g+). By Corollary 2.31(i) and Theo-

rem 2.30, the graph G2 belongs to C◦0(ga)∪C◦0(g+a) since G2 6∈ D, g+(G2) = 1, and g+ bounds

all the other parameters. Thus (ii) is true.

For (iii), suppose that G2 6∈ C◦0(g+a). Thus G2 is not g+-tight by Theorem 2.30 and thus

H = G. Since G2 6∈ C◦0(g+) and G2 is g-tight, Theorem 2.29 gives that η(G1, G2) = 1 (as

H10 ⊆ C◦0(g+) by Lemma 2.20). We conclude that either θ(G1) = 1 or θ(G2) = 1 and thus

(iii) holds.

By Lemma 2.34(iii), we have that H = G+ if and only if η(G1, G2) = 2. Since ε+(G2) = 0,

we obtain that η(G1, G2) = 2 if and only if θ(G1) = θ(G2) = 1. By Lemma 2.33 and

Corollary 3.24, θ(G1) = θ(G2) = 1 if and only if G1 ∈ C◦0(g+)\C◦0(g) and G2 ∈ C◦0(g+a)\C◦0(ga).

We conclude that H = G if and only if G1 ∈ C◦0(g) or G2 ∈ C◦0(ga). This finishes the proof

of the theorem.

Corollary 2.36. There are 68 obstructions of connectivity 2 for the torus.

CHAPTER 2. TORUS 37

Proof. By Theorem 2.35, for each H ∈ Forb(S1) of connectivity 2, there exists a 2-vertex-

cut {x, y} and an xy-sum G with parts G1, G2 ∈ G◦xy satisfying (i)–(iii). Let us count the

number of graphs in Forb(S1) of connectivity 2 by counting the number of non-isomorphic

xy-sums satisfying (i)–(iii).

Let us first count the number of pairs G1 and G2 for which (i), (ii), and (iii) of Theo-

rem 2.35 hold. The graphs in T1 are disconnected so their 2-sum with G1 is not 2-connected.

The number of connected graphs in C◦0(ga) ∪ C◦0(g+a) is |T2 ∪ · · · ∪ T6| = 27 and the number

of graphs in C◦0(g+) is 3. Thus we have precisely 81 pairs satisfying (i) and (ii). However,

some of them do not satisfy (iii).

Let us consider property (iii). There is only a single graph in C◦0(g+) that has θ equal to

0 (Fig. 2.5c). By Theorem 3.22, there are precisely |T4| = 5 graphs in C◦0(ga) \ C◦0(g+a); they

all have θ equal to 0 by Corollary 3.24. Thus 5 pairs out of the total of 81 do not satisfy

(iii) of Theorem 2.35 giving the total of 76 pairs satisfying (i), (ii), and (iii).

For fixed graphs G1 and G2 in G◦xy, there are two different xy-sums with parts G1 and

G2 as there are two ways how to identify two graphs on two vertices. Since for each graph

in C◦0(g+) there is an automorphism exchanging the terminals, there is precisely one xy-sum

with parts G1 and G2 that satisfies (i).

Therefore, for each of the 76 pairs, there is a unique xy-sum satisfying (i)–(iii). By

Theorem 2.35, for each such xy-sum G, either G or G+ is an obstruction for the torus. Some

of the obtained obstructions are isomorphic though. Let G be an xy-sum of connected

graphs G1, G2 ∈ G◦xy and G′ be an x′y′-sum of connected graphs G′1, G′2 ∈ G◦xy such that

both G and G′ satisfy (i)–(iii), let H ∈ {G, G+} and H ′ ∈ {G′, G′+} as given by the theorem,

and suppose there is an isomorphism ψ of H and H ′. If ψ({x, y}) 6= {x′, y′}, then ψ({x, y})is a 2-vertex-cut in G′. It is not hard to see that G′ has another 2-vertex cut only if G′2 ∈ T5.

We can see that the preimage of ψ of one part of G′ is a graph in C◦0(g+) \ C◦0(g). Therefore,

G1 ∈ C◦0(g+) \C◦0(g) and G2 ∈ T5 ⊆ C◦0(g+a) \C◦0(ga). Thus H = G+. But ψ(x) is not adjacent

to ψ(y), a contradiction.

We may assume now that ψ({x, y}) = {x′, y′}. If ψ(V (G1)) = V (G′1), then G1∼= G′1 and

G2∼= G′2 as argued above. Thus ψ(V (G1)) 6= V (G′1). It is not hard to check that only the

graphs in T2 have a subgraph isomorphic to a graph in C◦0(g+). There are 18 pairs G1, G2

such that G1 ∈ C◦0(g+) and G2 ∈ T2; but there are precisely 10 non-isomorphic obstructions

for the torus obtained from these 18 pairs. We conclude that there are 68 non-isomorphic

obstructions of connectivity 2 for the torus.

CHAPTER 2. TORUS 38

2.7 Open problems

The following questions remain unanswered:

(i) Do hoppers exist? If they do, what is the smallest k such that the class H0k (H1

k, or

H2k) is non-empty?

(ii) Is there a graph G ∈ C◦(g) with θ(G) = 1? In other words, can the graphs G and G+

be both obstructions for an orientable surface? What is the smallest k such that this

is the case for a graph of genus k?

(iii) What is the smallest k(r) such that there exists an r-connected obstruction G of genus

k with a pair of vertices x, y such that G is not xy-alternating. For example, k(0) = 2.

We do not know the value k(r) for r > 0.

Chapter 3

Graphs Critical for the Alternating

Genus

In this chapter we study the classes C◦0(ga) and C◦0(g+a). For k ≥ 0, let Ak(ga) be the subclass

of Gxy such that ga(G) ≤ k for each G ∈ Ak(ga). The class C0(ga) is closely related to C◦0(ga)

and C◦0(g+a) (see Lemma 2.16) and it has the following property: By Lemma 2.21, graph G

belongs to A0(ga) if and only if G has no graph in C0(ga) as a minor. By Lemma 2.16,

G ∈ A0(ga) if and only if G has no graph in C◦0(ga) as a minor and either xy 6∈ E(G) or G

has no graph in C◦0(g+a) as a minor. Note that each vertex of a graph in C0(ga) has degree

at least 3 unless it is a terminal.

The classes C◦0(ga) and C◦0(g+a) will be described using six subclasses T1, . . . , T6 of G◦xy. To

depict some of those classes, we use XY-labelled graphs defined in Section 3.3. Let T1 be the

class of graphs that contains each G ∈ G◦xy such that G is isomorphic to a Kuratowski graph

plus one or two isolated vertices that are terminals in G, T2 the class of graphs shown in

Fig. 3.4, T3 the class of graphs corresponding to the graphs in Fig. 3.8, T4 the class of graphs

corresponding to the graphs in Fig. 3.7, T5 the class of graphs depicted in Fig. 3.2, and T6

the class of graphs corresponding to the graphs in Fig. 3.6. Let T ′5 and T ′6 be the subclasses

of Gxy obtained from T5 and T6, respectively, by adding the edge xy to each graph.

Let T = T1∪T2∪T3∪T4∪T ′5 ∪T ′6 be a subclass of Gxy. Most of this chapter is devoted to

proving that C0(ga) = T and is organized as follows. In Section 3.1, we study the class C(ga)in general. The rest of the chapter is focused on the class C0(ga). A basic classification of

C0(ga) is shown in Section 3.2 and the complete list of C0(ga) is provided in the subsequent

39

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 40

chapters. The chapter is concluded in Section 3.6 where the classes C0(ga), C◦0(ga), and

C◦0(g+a) are determined.

3.1 General properties

In this section we present some general results about graphs in C(ga). In the sequel, G∗

denotes the graph obtained from G ∈ Gxy as the xy-sum of G and K5 − xy (Fig. 2.5a). We

will use a characterization of xy-alternating graphs by Decker et al. [12], that a graph G

with terminals x and y is xy-alternating if and only if g(G∗) = g(G). Note that both K5

and K3,3 are xy-alternating on the torus for any pair of vertices x and y (see Fig. 2.1).

For a graph G and a vertex x of G, the graph G′ is obtained by splitting G at x if x is

replaced by two adjacent vertices x1 and x2 and edges incident with x in G are distributed

arbitrarily to x1 and x2 in G′. By doing the same except that x1 and x2 are non-adjacent,

a resulting graph G′ is said to be obtained by cutting of G at x.

Suppose that a graph G is embedded in some surface S. Let γ be a simple closed curve

in S that intersects the embedded graph G only at vertices of G. The number of vertices in

γ ∩V (G) is called the width of γ (with respect to the embedded graph). If γ intersects G at

a vertex z, then it separates the edges incident with z into two parts, γ-sides at z, according

to their appearance in the local rotation around z. The graph obtained by cutting G at

each vertex v in γ ∩ V (G) using the γ-sides to partition the edges is said to be obtained by

cutting G along γ. The curve γ also induces the cutting of the surface S along γ, and the cut

graph is embedded in the cut surface. We say that γ is one-sided if any small neighborhood

of γ in S is connected. A curve is orientizing for a Π-embedded graph G if cutting G along

γ yields an orientable embedding of the resulting graph using the embedding induced by Π.

The orientizing face-width of G is the minimum width of an orientizing curve.

The next lemma outlines three characterizations of Ak(ga).

Lemma 3.1. Let G ∈ Gxy. If G does not embed into Sk−1, then the following statements

are equivalent:

(i) G is in Ak(ga).

(ii) G has an embedding Π into N2k−1 with an orientizing one-sided simple closed curve γ

of width 2 going through x and y.

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 41

(iii) G can be cut at x and y so that the resulting graph embeds into Sk−1 with x1, y1, x2

and y2 appearing on a common face (in the stated order).

(iv) G∗ embeds into Sk.

The proof of Lemma 3.1 uses the following result by Archdeacon and Huneke [2].

Lemma 3.2. Let G be a Π-embedded graph and W a Π-facial walk. If two vertices, x and

y, appear twice in W in the alternating order x, y, x, y, then there exists an embedding Π′

of G of Euler genus g(Π)− 1 such that every Π-facial walk is Π′-facial except for W which

turns into two Π′-facial walks W1 and W2, each of which contains both x and y. Moreover,

the curve γ passing through x and y and the faces W1 and W2 is one-sided in Π′ and the

signatures of edges in Π′ differ from Π only by switching the signatures of a γ-side at x and

a γ-side at y.

Proof of Lemma 3.1. The equivalence of (i) and (iv) was proven by Decker et al. [12].

(i)⇒(ii): Since G does not embed into Sk−1, it Π-embeds into Sk with x and y alternating

in a Π-face W . By Lemma 3.2, there is an embedding Π′ of Euler genus 2k − 1 with two

Π′-faces W1 and W2, both containing x and y. The curve γ obtained by connecting vertices

x and y in both faces W1 and W2 is the sought one-sided curve of width 2. Since the

signatures of edges in Π are positive, the edges of negative signature in Π′ form two γ-sides

of x and y (respectively). Thus cutting G along γ yields an orientable embedding and γ is

orientizing.

(ii)⇒(iii): Cutting along the one-sided orientizing curve γ yields an orientable embedding

Π of genus k − 1. Since γ is one-sided, the vertices obtained by cutting G along y lie on a

common face in the interlaced order.

(iii)⇒(i): Take an embedding Π of the resulting graph G′ into Sk−1 with x1, y1, x2, y2

on a common face W . Let G′′ = G′ + x1x2 + y1y2. We extend Π to an embedding Π′ of G′′

into Sk by embedding the new edges into W (and adding a handle). The number of faces

of Π′ stays the same but the number of edges is increased by two. Thus g(Π′) = g(Π) + 1.

By contracting the edges x1x2 and y1y2, we obtain G and its xy-alternating embedding

in Sk.

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 42

3.2 Basic classification

To determine the class C0(ga), we aim to understand the class A0(ga). Lemma 3.1 gives the

following characterizations of A0(ga).

Corollary 3.3. Let G ∈ Gxy be nonplanar. The following statements are equivalent:

(i) G is in A0(ga).

(ii) G has an embedding Π into the projective plane of face-width 2 with a non-contractible

curve of width 2 going through x and y.

(iii) G can be cut at x and y so that the resulting graph is planar with x1, x2, y1 and y2

on a common face.

(iv) G∗ embeds into the torus.

By Corollary 3.3, a nonplanar graph G belongs to A0(ga) if and only if the vertices x

and y can be split so that the resulting graph is planar with the new vertices on a common

face. This implies that G/xy is planar.

Corollary 3.4. If G is a nonplanar graph in A0(ga), then G/xy is planar.

We will show below that, if G/xy is nonplanar, then there is a Kuratowski subgraph in

G with a K-graph disjoint from x and y. The following lemma by Juvan et al. [17] allows

us to choose a subgraph without local bridges provided that we have an almost 3-connected

graph. Let K be a subgraph of G. The graph G is 3-connected modulo K if for every vertex

set U ⊆ V (G) with at most 2 elements, every connected component of G − U contains a

branch vertex of K.

Lemma 3.5 (Juvan, Marincek, and Mohar [17]). Let K be a subgraph of a graph G. If G

is 3-connected modulo K, then G contains a subgraph K ′ such that

(i) K ′ is homeomorphic to K and has the same branch vertices as K.

(ii) For each branch e of K, the corresponding branch e′ of K ′ joins the same pair of

branch vertices as e and is contained in the union of e and all K-bridges that are local

on e.

(iii) K ′ has no local bridges.

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 43

x

y

w1

w2

x

y

w1

w2

Figure 3.1: A case in the proof of Lemma 3.6.

Now, we are ready to prove that if G/xy is nonplanar, then there is a Kuratowski

subgraph in G with a K-graph disjoint from x and y.

Lemma 3.6. Let G be a nonplanar graph and x, y ∈ V (G). If G/xy is nonplanar, then G

contains a K-graph disjoint from x and y.

Proof. Suppose that the conclusion of the lemma is false. Let G be a counterexample with

|V (G)|+ |E(G)| minimum. It is easy to see that G is connected. If G−x is nonplanar, then

by Theorem 1.3, G− x contains a Kuratowski graph K and thus K − y contains a K-graph

in G that is disjoint from x and y. Hence G− x is planar. Similarly, G− y is planar.

Let K be a Kuratowski subgraph in G/xy and L a K-graph contained in K − vxy. Let

Bx and By be the L-bridges of G containing x and y, respectively. Necessarily, Bx and By

are different L-bridges of G since otherwise L is a K-graph in G disjoint from x and y. We

aim to get rid of the local L-bridges by applying Lemma 3.5 but also preserve the property

that the graph is a K-graph in G/xy that is disjoint from x and y. In order to achieve that,

we consider the graph G0 = G−B◦x −B◦y −w1w2 in the case when L is isomorphic to K2,3,

w1, w2 are the vertices of degree 3 in L, and w1w2 ∈ E(G). Otherwise, let G0 = G−B◦x−B◦y .

If G0 is not 3-connected modulo L, then there is a (minimal) vertex set U with |U | ≤ 2

such that a U -bridge C does not contain any branch vertex (in C◦). If |U | ≤ 1, then C is

a block of G0. Since genus is additive over blocks (see [3]), the block C is planar and its

removal from G yields a subgraph of G that satisfies the assumptions of the lemma. The

block C is planar because G−x and G−y are planar and since genus is additive over blocks

(by Theorem 2.1), the removal of C yields a subgraph of G that satisfies the assumptions

of the lemma. This is a contradiction with the choice of G being minimal. Thus U contains

exactly two vertices, u and v, and there is a path in C that connects u and v. Let G′ be the

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 44

graph obtained from G by contracting C into a single edge uv. Since C does not contain

x and y, if C + uv is nonplanar, then C contains a K-graph disjoint from x and y in G.

Hence C + uv is planar and Lemma 1.6 gives that G′ is nonplanar. It is not difficult to

see that G′/xy is also nonplanar. By the choice of G, there is a K-graph L′ in G′ disjoint

from x and y. Since the edge uv in G′ can be replaced in G by a path in C, L′ induces in a

straightforward way a K-graph in G disjoint from x and y.

Therefore, we may assume that G0 is 3-connected modulo L. By Lemma 3.5, there exists

a subgraph L′ of G0 homeomorphic to L that has no local bridges, and has the same branch

vertices as K ′ and also satisfies property (ii) of Lemma 3.5. Note that, since K2,3 and K4

are uniquely embeddable in the plane, L′ has a unique planar embedding Π. Let B′x and

B′y be the L′-bridges in G containing x and y, respectively. By using (ii) of Lemma 3.5, it

is not difficult to check that L′ is still a K-graph in G/xy. It follows that B′x and B′y are

different L′-bridges in G.

Case 1: L′ is a subdivision of K4 or w1w2 6∈ E(G).

Since G−B′◦x and G−B′◦y are planar, each L′-bridge can be embedded into some Π-face.

Since only B′x and B′y can be local L′-bridges in G, each other L′-bridge in G embeds into

a unique Π-face. Since the vertices of the union of the attachments of B′x and B′y do not

lie on a single Π-face, the bridges B′x and B′y embed into different Π-faces. We conclude

that each L′-bridge in G can be assigned a Π-face such that all bridges assigned to a single

Π-face can be embedded there simultaneously. Hence G is planar — a contradiction.

Case 2: L is a subdivision of K2,3 and w1w2 ∈ E(G).

Consider the graph G′′ = G−w1w2. Since G′′ is a subgraph of G and G′′/xy is nonplanar,

G′′ is planar by the choice of G . Since the planar embedding of G′′ cannot be extended into

a planar embedding of G by adding the edge w1w2 into one of the three Π-faces, there are

three paths P1, P2, P3 that connect the three pairs of open branches of L′, respectively (see

Fig. 3.1). Let L′′ be the subgraph of G that consists of w1w2, the path Pi that is embedded

in the Π-face containing neither x nor y and the two branches of L′ that Pi connects to. It is

easy to see that L′′ forms a K-graph in G that is disjoint from x and y, a contradiction.

Lemma 3.6 leads to the following dichotomy of graphs is C0(ga).

Lemma 3.7. Let G ∈ C0(ga). Then one of the following is true:

(i) G ∈ T1,

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 45

x y x y x y

Figure 3.2: T5, splits of Kuratowski graphs which belong to C◦0(g+a) \ C◦0(ga)

XY

XY

X

Yx

y

Figure 3.3: An example of an XY-labelled graph and its corresponding graph in G◦xy.

(ii) G− xy ∈ T5, or

(iii) G/xy is planar.

Proof. Suppose that G does not satisfy (iii). By Lemma 3.6, there is a Kuratowski subgraph

K in G with a K-graph L disjoint from x and y. If there exists a minor-operation µ ∈M(G)

such that µG still contains a K-graph disjoint from x and y, then (µG)/xy is nonplanar

and thus µG 6∈ A0(ga) by Corollary 3.4. Hence E(G) = E(K). If x and y lie in different

components of G, then (i) holds. Otherwise, x and y lie in the unique L-bridge. Consider an

edge u1u2 ∈ E(K) and the graph G/u1u2. If u1 or u2 is not a terminal nor a branch-vertex

of K that lies in L, then G/u1u2 still contains a K-graph disjoint from x and y. We conclude

that u1 and u2 are either terminals or branch-vertices of K contained in L. Now it is easy

to see that G satisfies (ii).

3.3 XY-labelled graphs

To investigate graphs in G ∈ C0(ga) where G/xy is planar, we study the graph G − x − y.

Given a graph G ∈ G◦xy, let H be the graph G− x− y where a vertex of H is labelled X if

it is adjacent to x in G and it is labelled Y if it is adjacent to y in G (see Fig. 3.3). Thus

each vertex of H is given up to two labels. Let λ(v) denote the set of labels given to the

vertex v of H. A vertex v is labelled if λ(v) is non-empty. The graph H together with the

labels carries all information about G. We say that H is an XY-labelled graph corresponding

to G. Let LXY be the class of XY-labelled graphs corresponding to a graph G ∈ G◦xy such

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 46

that G/xy is planar. The notion of a minor of a graph is extended to XY-labelled graphs

naturally: an XY-labelled graph H1 is a minor of an XY-labelled graph H2 if the graph

with terminals corresponding to H1 is a minor of the graph with terminals corresponding to

H2. For example, the deletion of a label is a minor operation that corresponds to an edge

deletion and, when contracting an edge uv in an XY-labelled graph, the resulting vertex is

labelled by λ(u) ∪ λ(v).

Let H ∈ LXY and consider the multigraph H and the vertex vxy obtained by identifica-

tion of x and y in G (in contrast to the simple graph G/xy used in the previous sections).

Label each edge e of H incident to vxy by the label X (Y) if the edge was incident to x

(y) in G. Let Π be a planar embedding of H. The local rotation around vxy gives a cyclic

sequence S of labels that appear on the edges incident with vxy. Call S a label sequence

of H. A label transition in a label sequence is a pair of (cyclically) consecutive labels that

are different. The number of transitions τ(Q) of S is the number of label transitions in

S. In the case when S contains only two different labels, τ(Q) is a multiple of 2. Thus

we say that a label sequence S is k-alternating if τ(Q) = 2k. A planar embedding of H is

k-alternating if the induced label sequence is k-alternating and H is called k-alternating if

H admits a k-alternating embedding in the plane. Note that Corollary 3.3 implies that, if

H is 2-alternating, then the corresponding graph G is in A0(ga).

Let H ∈ LXY. When H is connected, a planar embedding of H induces a planar

embedding of H with a special face W in which vxy is embedded. Call the cyclic sequence

of vertices of W (with some possibly appearing more than once) a boundary of H. If H is

2-connected, then W is a cycle of H (see [20, Thm. 2.2.3]). To understand when a planar

embedding of H induces a 2-alternating label sequence, we study the possible boundaries

of H. If M is a block of H that is not an edge, then a boundary of H induces a boundary

cycle in M .

A sequence R = v1, . . . , vk of consecutive vertices on a boundary Q is called an X-block

in Q if no vertices in R except possibly the endvertices v1 and vk are labelled with Y. Define

a Y-block similarly.

The following lemma states the observation that, if two X-blocks contain all vertices

that are labelled X, then it is easy to construct a 2-alternating embedding of H. In this

case, we say that the labels X are covered by the two X-blocks.

Lemma 3.8. Let H ∈ LXY, Q a boundary of H, and A ∈ {X,Y}. If the A-labelled vertices

of H are covered by two A-blocks in Q, then H is 2-alternating.

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 47

x y x y x y x y x y x y

Figure 3.4: T2, the xy-sums of graphs in C◦0(g+) which belong to C◦0(ga) ∩ C◦0(g+a).

For A ∈ {X,Y}, an induced subgraph H ′ of H contains the label A if there is a vertex

in H ′ labelled A. Let S = A1 . . . Ak be a label sequence. Here we consider S as a linear

label sequence as opposed to cyclic. Let R be a subsequence of a boundary of H. We say

that R contains the label sequence S if there are distinct vertices v1, . . . , vk that appear in

R in this order (or the reverse order) and vi is labelled Ai for i = 1, . . . , k. We say that H ′

contains the label sequence S if for every boundary Q of H, the subsequence of Q induced

by V (H ′) contains the label sequence S. Let B be a block of H and v a vertex of B. We

say that label A is attached to B at v if either v is labelled A or there is a v-bridge in H

not containing B that contains A.

Lemma 3.9. Let H ∈ LXY such that at most four vertices of H have both labels X and Y.

If H is not 2-alternating, then H contains the label sequence XYXYXY.

Proof. Suppose that H is not 2-alternating and let Q be a boundary of H. Let R be a

subsequence of Q with no unlabelled vertices such that each labelled vertex appears in R

exactly once. A stronger claim is proved instead. If R does not contain the label sequence

XYXYXY, then the labels of vertices inR can be arranged in the order given byR to obtain a

2-alternating sequence of labels. Suppose that this is not true and choose a counter-example

R with minimum total number of labels.

Suppose there are cyclically consecutive vertices u and v in R such that both u and v

have label A and v has only one label. By deleting A from u we obtain a sequence R′ with

smaller total number of labels. By the construction of R′, R′ does not contain the label

sequence XYXYXY. Thus there is a 2-alternating label sequence S′ of labels in R′. By

inserting the label A before the occurrence of A at v, we obtain a valid 2-alternating label

sequence for R. Therefore, every two consecutive vertices in R have either distinct labels or

both labels.

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 48

Triangle

XY

XY

XY

Square

X

Y

X

Y

Claw

XY

XY

XY

Figure 3.5: The XY-labelled representation of C◦0(g+).

Two cases remain: Either R contains at most four labelled vertices, all with both labels,

or there are at most four vertices that have alternating labels (six vertices give the label

sequence XYXYXY and five vertices are not possible because of parity). In both cases, we

see immediately that the labels in R can be arranged into a 2-alternating label sequence.

For graphs in C0(ga), Lemma 1.6 gives the following result.

Corollary 3.10. Let G ∈ C0(ga) and let {u, v} be a 2-vertex-cut in G. If B is a non-trivial

uv-bridge such that B + uv is planar, then B◦ contains a terminal.

The following lemma describes the structure of a graphs in C0(ga) when the XY-labelled

graph is disconnected.

Lemma 3.11. Let H ∈ LXY such that H is not 2-alternating and let G be the graph

corresponding to H. If H is disconnected, then G has a graph in T2 as a minor.

Proof. Since H is disconnected, G has at least two non-trivial xy-bridges B1 and B2. Since

neither B◦1 nor B◦2 contains a terminal, Corollary 3.10 gives that both B1 + xy and B2 + xy

are nonplanar. Hence B1 and B2 contain one of the graphs in C◦0(g+) as a minor. We

conclude that G contains one of the graphs in T2 as a minor.

Fig. 3.5 shows the XY-labelled representation of graphs in C◦0(g+).

3.4 Connectivity 2

This section is devoted to the proof of the following lemma characterizing graphs in C0(ga)

that correspond to a 2-connected graph in LXY.

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 49

Pentagon

XY

XY

XY

XY

XY

Hexagon

X

YX

Y

X Y

Figure 3.6: The XY-labelled representation of T6 ⊆ C◦0(g+a) \ C◦0(ga).

Lemma 3.12. Let H ∈ LXY be 2-connected and not 2-alternating and let G be the graph

corresponding to H. Then G has a graph in T6 as a minor. Furthermore, if G is nonplanar,

then G has one of the graphs in T4 \ {Lollipop} as a minor.

First, we derive two lemmas that will be used in the proof of Lemma 3.12.

Lemma 3.13. Let G be a graph that consists of a cycle C and planar C-bridges B1, B2

such that all other C-bridges are planar and avoid each other. If G is nonplanar, then there

is a C-bridge B (different from B1 and B2) such that B, B1, and B2 all pairwise overlap.

Proof. Let B be the set of C-bridges in G different from B1 and B2. Since the bridges in Bavoid each other, B forms an independent set in O(G,C). Since G is nonplanar, Theorem 1.1

asserts that O(G,C) is non-bipartite and thus contains an odd cycle. Since every edge in

O(G,C) is incident with B1 or B2, this odd cycle is a triangle that consists of B1, B2 and

a bridge B ∈ B.

Lemma 3.14. Let H be an XY-labelled planar graph that consists of an XY-labelled cycle

C and a C-bridge B. Let C[w1, w2] be a segment of C that contains all attachments of B.

If C contains all labels of H and the graph with terminals corresponding to H is nonplanar,

then C(w1, w2) contains both labels.

Proof. Let G ∈ G◦xy be the graph corresponding to H. Let Bx and By be the C-bridges that

contain x and y, respectively. Since C contains all labels of H, Bx and By are stars attached

only to C. By Lemma 3.13, the bridges B, Bx and By pairwise overlap. Theorem 1.4 implies

that, for each z ∈ {x, y}, either

(i) there are disjoint crossing paths P1 in B and P2 in Bz, or

(ii) the bridges B and Bz have three vertices of attachment in common.

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 50

Rocket

XY

XY

XY

XY

XY

Lollipop

XY

XY

XY

XY

XY

Bullet

XY

YX

Y

X Y

Frog

X

YX

Y

X Y

Hive

X

YX

Y

X Y

Figure 3.7: The XY-labelled representation of T4 = C◦0(ga) \ C◦0(g+a). The underlined labels

are used in the proof of Lemma 3.23.

Let Z be the label corresponding to the vertex z. When (i) holds, C(w1, w2) contains one

of the endvertices of P2 and thus contains the label Z. When (ii) holds, each attachment of

B is labelled Z. Since C(w1, w2) contains at least one of the attachments of B, C(w1, w2)

contains the label Z. Therefore, C(w1, w2) contains both labels X and Y as claimed.

We conclude this section with the proof of Lemma 3.12.

Proof of Lemma 3.12. Let C be a boundary cycle of H and Π the corresponding planar

embedding of H. By Lemma 3.9, either C contains the label sequence XYXYXY, and then

H has Hexagon as a minor, or there are five vertices in C with both labels, and then H

has Pentagon as a minor.

Therefore, we may assume that G is nonplanar. Let us consider the C-bridges Bx and

By in G that are the stars with centers x and y, respectively. We may assume that, in Π,

C is the boundary of the outer face. By Lemma 3.13, there is a C-bridge B such that B,

Bx, and By pairwise overlap.

Let us first consider the case when C does not contain the label sequence XYXYXY.

By Lemma 3.9, C contains five vertices with both labels. Let v1, . . . , v5 be the vertices with

both labels. We may assume by symmetry that an attachment of B lies in C(v1, v3). If

there is an attachment of B in the segment C(v3, v1), then H has Rocket as a minor.

Otherwise, all attachments of B are in the segment C[v1, v3]. Let S be the support of B

in C[v1, v3]. By Lemma 3.14, the segment S (excluding the endvertices of S) contains both

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 51

labels. Thus H has Rocket as a minor.

Now, assume that C contains the label sequence XYXYXY and let v1, . . . , v6 be the

vertices manifesting that (so X ∈ λ(v1), Y ∈ λ(v2), etc.). Let w1, . . . , wk be the attachments

of B. Note that k ≥ 2. By symmetry, we may assume that w1 lies in the segment C(v1, v3).

If all attachments of B lie in C[v1, v3], then the support S of B in C[v1, v3] (excluding

the endvertices of S) contains both labels by Lemma 3.14. Thus H has Bullet as a minor.

Hence we may assume that not all attachments of B are in C[v1, v3] and similarly in C[v2, v4]

and so on. If there is an attachment of B in the segment C(v4, v6), then H has Frog as a

minor. Hence we may assume that all attachments of B lie in the segment C[v6, v4].

By using reflection symmetry exchanging v1, v3 and v4, v6, since not all attachments of

B are in C[v1, v3], there is an attachment w2 of B in the segment C(v3, v4]. By the same

argument as above, there is no attachment of B in C(v6, v2). Since not all attachments of

B are in C[v2, v4], the vertex v6 is an attachment of B. We conclude that H has Hive as a

minor.

3.5 Connectivity 1

In this section, we describe all obstructions in C0(ga) that correspond to a graph in LXY of

connectivity 1.

Lemma 3.15. Let H ∈ LXY such that H has connectivity 1 and is not 2-alternating and

let G be the graph corresponding to H. Then G has one of the graphs in T3 ∪ {Lollipop}as a minor.

For graphs in C0(ga), Lemma 1.5 has the following consequence.

Corollary 3.16. Let G ∈ C0(ga) and uvw be a triangle in G. If u has degree at most 3 in

G, then u is a terminal.

Proof. Since G−vw ∈ A0(ga), either G−vw is planar or G−vw is xy-alternating on S1. By

Lemma 1.5, the first outcome is not possible since then G would be planar. In the second

case, Lemma 1.5 shows that the xy-alternating embedding of G− vw can be extended into

an embedding of G in S1 by embedding vw along the path vuw. This extension would be

xy-alternating if u 6∈ {x, y}. Thus, u is one of the terminals.

The next lemma will be used throughout the rest of the chapter.

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 52

Lemma 3.17. Let H be an XY-labelled graph that has distinct blocks B1 and B2. Suppose

that each of B1 and B2 contains both labels X and Y on vertices that do not belong to another

block. Let G be the graph with terminals corresponding to H. If H is not 1-alternating, then

G is nonplanar.

Proof. Suppose for contradiction that G is planar and take a planar embedding Π of G. If

x and y are cofacial in Π, then Π gives a 1-alternating embedding of H. If x and y are not

cofacial in Π, then there is a cycle C in H that separates x and y (since x and y lie inside

different faces of the induced embedding of H). Since C is a cycle of H, it intersects either

B1 or B2 in at most one vertex. Say, B1 shares at most one vertex with C and is embedded

on the other side of C than x is. By assumption, there is a vertex v ∈ V (B1) \ V (C) that is

labelled X. Clearly, v and x are not cofacial in Π since they are separated by C. But v and

x are adjacent and thus cofacial in Π, a contradiction.

Let C be a block in a graph G. The C-bridge set Bv at a vertex v of C is the union of

all C-bridges in G that are attached to v. The following lemma asserts several properties of

H and its labels and it is used to classify the graphs of connectivity 1 in C0(ga).

Lemma 3.18. Let H ∈ LXY be a graph of connectivity 1 and let G be the graph correspond-

ing to H. If G ∈ C0(ga), then the following statements hold:

(N1) Vertices of degree at most 2 in H are labelled. Leaves in H have both labels.

(N2) If B is an endblock of H, and v is a cutvertex that separates B from the rest of H,

then the graph B − v contains both labels.

(N3) Let M be a block of H that is not an edge and C a boundary cycle of M . Let B be

the subgraph of M that consists of C-bridges in M . If B is non-empty, then H − B◦

is not 2-alternating.

(N4) Each block of H is either an edge or a cycle.

(N5) Let u be a vertex of degree 2 in H. If u has only one label, then the neighbors of u

are not labelled by λ(u). In particular, if P is a path in H such that each vertex of P

has degree 2 in H, then either each vertex of P has both labels or each vertex of P has

precisely one label that is different from the labels of its neighbors.

(N6) The neighbor of a leaf in H is unlabelled.

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 53

(N7) Let C be a cycle of H and T a C-bridge set that is a tree. If H consists of at least

three blocks, then T contains at least two leaves of H.

(N8) Let B be a block of H that is a triangle and v a vertex of B. If v is not a cutvertex,

then it has both labels. Otherwise, both labels are attached to B at v.

Proof. Each property is proved separately.

(N1): Vertex of degree 2 in H with no label would be a vertex of degree 2 in G. Similarly,

a vertex of degree 1 with at most one label would be a vertex of degree at most 2 in G.

(N2): Let B be an endblock of H and v ∈ V (B) the cutvertex that separates B from the rest

of the graph. If B is an edge, then the result follows from (N1). Suppose for contradiction

that B − v does not contain the label Y. Since G/xy is planar, B is either a planar block

of G or B is in an xv-bridge C of G such that C + xv is planar. Corollary 3.10 asserts that

this cannot happen in G.

(N3): Suppose B is non-empty and Π is a 2-alternating embedding of H −B◦ in the plane.

Suppose that there is an edge e of H −B◦ with one end v in C. By construction of H −B◦,e lies in a different v-block B of H than C. By (N2), there is a vertex u in B labelled X.

Thus there is a path P in H −B◦ that connects vxy and v and is internally disjoint from

C. It follows that e is embedded on the same side of C in Π as x and y. We conclude that

C is a Π-face. By construction of C, Π can be extended to a 2-alternating embedding of H

by embedding B inside C — a contradiction.

(N4): Let M be a block of H that is neither a cycle nor an edge. Let C be a boundary

cycle of M and B the subgraph that consists of C-bridges in M . By (N3), G − B◦ is not

xy-alternating on the torus. By (N2), H − B◦ contains two endblocks that contain both

labels. By Lemma 3.17, G−B◦ is nonplanar, a contradiction with G−B◦ ∈ A0(ga).

(N5): By (N1), u is labelled, say by X. If v is a neighbor of u with label X, then u is a

vertex of degree 3 in the triangle uvx which is not possible by Corollary 3.16 unless u is also

labelled Y.

(N6): Let v be a leaf and u its neighbor. If u is labelled, say with label X, then v is a vertex

of degree 3 in the triangle vxu which is not possible by Corollary 3.16.

(N7): Let C be a cycle and T be a C-bridge set that is a tree. Assume that H has at least

3 blocks and that T contains only one leaf. We see that T is a path and, by (N6) and (N1),

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 54

Star

XY

XY

XY

XY

XY

Alien

XY

XY

XYXY

XY

Ribbon

X

YXY

XY XY

XY

Saddle

XYXY

XY XY

XY

Four

XY

XY

XYXY

XY

Five

Y

X

Y

XXY

XY

Human

X

Y

X

YXY

XY

Bowtie

XY

XY

XYXY

XY

Doll

X

XY

XY

X

Y

Y

Pinch

X

Y

X

X

Y

Y

Extra

X

Y

X

Y

X

Y

Y

Figure 3.8: The XY-labelled representation of T3 ⊆ C◦0(ga) ∩ C◦0(g+a). For each white vertex

v ∈ V (G), we have g(G− v) = 1.

it is a path of length 1. Contract T to C to get H ′. Let G′ be the graph corresponding to

H ′. By the choice of G, G′ is either xy-alternating on the torus or planar. Since H either

contains 3 endblocks or two disjoint endblocks, if G′ is not xy-alternating on the torus, then

Lemma 3.17 gives that G′ is nonplanar. Hence G′ is xy-alternating on the torus. Let Π be a

2-alternating embedding of H ′ in the plane. Uncontract T to get a 2-alternating embedding

of H — a contradiction.

(N8): Let v be a vertex in a triangle C with at most one label. If v is not a cutvertex, then

v is has degree at most 3 in G. By Corollary 3.16, this is a contradiction. If v is a cutvertex,

then there is a v-bridge B′ that does not contain C. Since B′ contains an endblock of H,

(N2) implies that B′ contains both labels. These labels are attached to B at v.

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 55

We use the structural properties from Lemma 3.18 to prove Lemma 3.15.

Proof of Lemma 3.15. Let G and H be as in the statement of the lemma. Our goal is to

show that H has one of the graphs from T3 or T4 as a minor.

If H has at least five leaves, then all leaves are labelled X and Y, by (N1). Since H is

connected, H has Star as a minor. We assume henceforth that H has at most four leaves.

By (N4), every block of H that is not an edge is a cycle. We split the discussion according

to the number of cycles in H.

Case 1: H is acyclic.

Suppose H has k leaves w1, . . . , wk, where k ≤ 4. Let u1, . . . , uk be their neighbors

(possibly not distinct). By (N6) and (N1), vertices ui (where i = 1, . . . , k) have no labels

and are of degree at least 3. By a counting argument, there are at most two such vertices

in H. If there is only one vertex u of degree at least 3, H is a star with center u and thus

H is a proper minor of Star and hence G is in A0(ga). Thus, there are two of them, say

u1 and u2, and they are connected by a path P . If P contains both labels X and Y, then

H has Saddle as a minor. If P contains at most one of the labels, say X, then the two

pairs of leaves are covered by two Y-blocks and thus G is in A0(ga) by Lemma 3.8 — a

contradiction.

Case 2: H has precisely one cycle C.

Since C is the only cycle in H, every C-bridge is a tree attached to a vertex of C. The

proof is split according to the number of leaves of H. Note that H has at least one leaf

since H is not 2-connected.

Subcase i: H has precisely four leaves.

If C is an endblock, then a single C-bridge set Bv contains all four leaves w1, . . . , w4.

By (N2), C − v contains both labels. Therefore, H has Star as a minor.

Otherwise, by (N7), there are precisely two non-trivial C-bridge sets Bv1 and Bv2 , and

each contains two leaves. Hence each of Bv1−v1 and Bv2−v2 contains at most one vertex of

degree 3 in H. When Bv1−v1 contains a vertex of degree 3, let u1 be this vertex. Otherwise,

let u1 = v1. Define u2 similarly. Note that u1 and u2 are unlabelled by (N6) and (N1). If

there is a path P in H connecting u1 and u2 and both labels X and Y appear on P , then H

has Saddle as a minor. Let P1 and P2 be the two paths in C connecting v1 and v2. If P1

contains X and P2 contains Y (or vice versa), then H has Ribbon (or Saddle) as a minor.

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 56

Otherwise, there is a label missing from H − {wi : i = 1, . . . , 4}, say X, so the leaves are

covered by two X-blocks. Lemma 3.8 implies that G ∈ A0(ga), a contradiction.

Subcase ii: H has precisely three leaves.

By (N7), there is a single C-bridge set Bv that contains all three leaves. Suppose C is

a triangle. By (N8), both vertices of C different from v have both labels and H contains

Star as a minor.

Suppose C has length at least 4. By (N5), C − v contains the label sequence XYX or

YXY. Thus H has Star as a minor.

Subcase iii: H has precisely two leaves.

By (N7), there is a single C-bridge set Bv that contains both leaves. Let u be a vertex

of degree 3 in Bv − v if there is one and let u = v otherwise. Let P be the path from u to

v, possibly of zero length.

Suppose C is a triangle. Again by (N8), both vertices of C different from v have both

labels. If P contains both labels, then H has Alien as a minor (by (N6)). Thus P contains

at most one label, say X, and then labels Y are covered by two Y-blocks, one at the leaves

and one on the triangle. By Lemma 3.8, G ∈ A0(ga), a contradiction.

Suppose C has length at least 4. If all vertices in C − v have both labels, then H has

Four as a minor. If C−v contains the label sequence XYXY, then H has Five as a minor.

Otherwise, (N5) implies that C has length 4 and C − v form the label sequence YXY or

XYX, say the former. If P contains X, then H has Human as a minor. Otherwise, the

labels X are covered by two X-blocks, one at the leaves and one covering the label X at C

— a contradiction by Lemma 3.8.

Subcase iv: H has precisely 1 leaf.

Let w be this leaf and u its neighbor. By (N6) and (N1), u is unlabelled vertex of degree

at least 3 and thus lies on C. If C has length at most 5, then H contains five vertices with

both labels, by Lemma 3.9. Thus H is isomorphic to Lollipop. If C has length at least 6,

then it follows from (N5) that G has the graph corresponding to Hexagon plus the edge

xy as a proper minor.

Case 3: H has (at least) two cycles, C1 and C2.

Pick C1 and C2 such that, first, the distance between them is maximal and, second, their

size is maximal. By (N4), C1 and C2 are blocks of H that share at most one vertex. Let

P be a shortest path (possibly of zero length) joining vertices v1 ∈ V (C1) and v2 ∈ V (C2).

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 57

Note that by the choice of C1 and C2, all C1-bridges attached to C1− v1 and all C2-bridges

attached to C2 − v2 are trees.

Subcase i: C1 and C2 are triangles.

Suppose there is more than one C1-bridge at v1 and let B be a C1-bridge at v1 not

containing P . By (N2), B contains both labels. By (N8), all vertices of C1− v1 and C2− v2

have both labels attached. Thus H has Star as a minor. So we may assume that there is

only one C1-bridge attached at v1. Similarly, there is only one C2-bridge attached at v2.

If there is a C1-bridge attached to a vertex v of C1 − v1, then the C1-bridge set at v is

a tree containing at least two leaves by (N7). This implies that H has Star as a minor.

Thus there are no C1-bridges attached to C1 − v1. The same holds for C2 by symmetry.

If the component M of H − E(C1) − E(C2) containing P has both labels, then H has

Bowtie as a minor. Suppose to the contrary that M has at most one label, say X. Since

there are no other bridges attached to C1 and C2, the Y-labelled vertices of H are covered

by two Y-blocks, a contradiction by Lemma 3.8.

Subcase ii: C1 is a triangle and C2 has length at least 4.

If H contains four leaves, then it is not difficult to check that H has Star as a minor.

Hence there is at most one non-trivial bridge set attached to C1 − v1 or C2 − v2 (by (N7)).

Suppose that there is a C2-bridge set B attached to a vertex v in C2−v2. By (N7), B contains

at least two leaves. If B contains three leaves, then H has Star as minor. Therefore, B has

precisely two leaves w1, w2. Let M be the component of H − E(C1) − w1 − w2 containing

P . By using (N6), it is easy to see that, if M contains both labels, then H has Alien as

a minor. Otherwise, M has at most one label, say X. Thus labels Y are covered by two

Y-blocks, one at C1 − v1 and the other at w1, w2. A contradiction by Lemma 3.8.

Therefore, there is no C2-bridge attached to C2 − v2. By (N5), C2 − v2 either contains

the sequence YXY or XYX, say the former. Suppose there is a C1-bridge B attached at

C1 − v1. By (N7), B has at least two leaves. Hence H has Star as a minor. Therefore,

there is no C1-bridge attached at C1 − v1 and both vertices in C1 − v1 have both labels.

Let M be the component of H − E(C1) − E(C2) containing P . If M contains X, then

H has Doll as a minor. Hence M contains at most one label, Y. If C2 has length at least

5, then C2 − v2 contains the label sequence XYXY by (N5). It follows that H has Five as

a minor. Thus C2 has length 4. If all vertices in C2 − v2 contain both labels, then H has

Four as a minor. Otherwise, the labels X at C2−v2 can be covered by an X-block. Since all

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 58

other labels X are at C1 − v1 covered by one X-block, Lemma 3.8 implies that G ∈ A0(ga),

a contradiction.

Subcase iii: Both C1 and C2 have length at least 4.

By (N7), every bridge set attached to C1 − v1 and C2 − v2 contains at least two leaves.

Suppose there are non-trivial bridge sets B1 attached to C1−v1 and B2 attached to C2−v2,

respectively. Since H contains at most four leaves, B1 contains two leaves w1, w2 and B2

contains two leaves w3, w4. If M = H −{wi : i = 1, . . . , 4} contains both labels, then H has

Saddle as a minor. Otherwise, M has at most one label, say X. Hence all Y labels are at

the leaves and can be covered by two Y-blocks. A contradiction by Lemma 3.8. If there are

two non-trivial bridge sets B1, B2 attached to one of the cycles, say to C1, then B1 and B2

contain together four leaves. By (N2), there are both labels attached to a vertex of C2− v2.

Hence H has Star as a minor.

Therefore, there is at most one non-trivial bridge set attached to C1 − v1 and C2 − v2.

Suppose there is a C1-bridge set B attached to a vertex v in C1 − v1. By (N5), C2 − v2

contains the label sequence YXY or XYX, say the former. If B contains at least three

leaves, then H has Star as a minor. By (N7), B has precisely two leaves w1, w2. If C2 has

length at least 5, then C2 contains the sequence XYXY, by (N5). Hence H has Five as

a minor. If C2 contains three vertices with both labels, then H has Four as a minor. If

H − w1 − w2 − (C2 − v2) contains label X, then H has Human as a minor. Otherwise, the

X labels at C2 − v2 can be covered by a single X block and all other X labels are at w1, w2

which are covered by a second X block. By Lemma 3.8, H is 2-alternating, a contradiction.

By symmetry of C1 and C2, we conclude that there are no non-trivial bridge sets attached

to C1 − v1 and C2 − v2. By (N5), C2 − v2 contains the label sequence YXY or XYX, say

the former. By (N5), C1− v1 contains the label sequence YXY or XYX. If C1− v1 contains

the sequence XYX, then H has Pinch as a minor. Thus C1 − v1 contains the sequence

YXY. If C2 − v2 contains the sequence XYX, then H has Pinch as a minor. Let M be

the component of H −E(C1)−E(C2) that contains P . If M contains label X, then H has

Extra as a minor. Otherwise, the labels X can be covered by two X-blocks, one at C1− v1

and one at C2 − v2. A contradiction by Lemma 3.8.

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 59

3.6 The class C0(ga)

In this section, we determine the classes C◦0(ga) and C◦0(g+a). Let us start with a lemma

giving another characterization of C0(ga).

Lemma 3.19. Let C ⊆ Gxy be a class of graphs such that each graph not in A0(ga) has a

graph in C as a minor and no graph in C is a minor of another graph in C. If C∩A0(ga) = ∅,then C0(ga) = C.

Proof. Let G ∈ C0(ga). Since G 6∈ A0(ga), there is a graph G′ ∈ C that is a minor of G.

Since each proper minor of G belongs to A0(ga), we conclude that G ∈ C. Suppose there

exists a graph G′ ∈ C \ C0(ga). Since G′ 6∈ A0(ga), there is a graph G′′ ∈ C0(ga) that is a

minor of G′. Thus G′′ ∈ C which yields a contradiction.

We are now able to determine the class C0(ga).

Theorem 3.20. Let G ∈ Gxy. Then G ∈ C0(ga) if and only if G ∈ T .

Proof. By Lemma 3.19, it is sufficient to prove that (i) each graph G 6∈ A0(ga) has a minor

in T , (ii) no graph in T has a proper minor in T , and (iii) T ∩ A0(ga) = ∅. Let us start

with (i). Let G 6∈ A0(ga). We may assume that G ∈ C0(ga). By Lemma 3.7, either G ∈ T1,

G ∈ T ′5 , or G/xy is planar. Thus we may assume that G/xy is planar. Let H be the

XY-labelled graph corresponding to G−xy. Note that G is planar only if xy ∈ E(G). Since

G 6∈ A0(ga), the graph H is not 2-alternating. If H is disconnected, then G has a graph

in T2 as a minor by Lemma 3.11. If H has connectivity 1, then G has a graph in T3 ∪ T4

as a minor by Lemma 3.15. Otherwise, H is 2-connected. If G is planar, then G ∈ T ′6 by

Lemma 3.12. If G is nonplanar, then G ∈ T4 by Lemma 3.12. This shows (i).

By Corollary 3.3, to show (ii) and (iii), it is sufficient to prove that for each G ∈ T ,

the graph G∗ does not embed into the torus but, for each minor-operation µ ∈ M(G), the

graph µG∗ does embed into the torus. This can be checked by computer. Equivalently, (ii)

and (iii) can be checked by hand. The details are not included in this thesis.

The classes T1, . . . , T6 lie in C◦0(g+a) ∪ C◦0(ga). More precise membership as depicted in

Fig. 3.9 is proven below. We shall use the following observation.

Lemma 3.21. Let G ∈ Gxy, P a minor-monotone graph parameter, and v ∈ V (G) \ {x, y}.If P(G− v) = P(G), then P(µG) = P(G) for each µ = (uv, ·) ∈M(G).

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 60

C◦

0 (ga) C◦

0 (g+a)

T4 T1 ∪ T2 ∪ T3 T5 ∪ T6

Figure 3.9: Venn diagram of critical classes (for alternating genus) for the torus.

Proof. Let µ = (uv, ·) ∈ M(G). Since G − v is a minor of µG and P is minor-monotone,

P(G) ≥ P(µG) ≥ P(G− v) = P(G).

Lemma 3.21 can be used to prove that ∆1(g) = ∅ if we can find a vertex cover U

of G such that g(G − v) = g(G) for each v ∈ U . We shall use this idea to prove that

T3 ⊆ C◦0(g+a). In order to determine if a graph G ∈ C◦0(ga) also belongs to C◦0(g+

a) we can

either use Lemma 2.17 or note that, since g+a(G) ≥ ga(G) and g+

a is minor-monotone by

Lemma 2.3, each graph G ∈ C◦0(ga) contains a graph in C◦0(g+a) as a minor.

Theorem 3.22. C◦0(ga) ∪ C◦0(g+a) = T1 ∪ · · · ∪ T6.

Furthermore,

(i) C◦0(ga) ∩ C◦0(g+a) = T1 ∪ T2 ∪ T3,

(ii) C◦0(ga) \ C◦0(g+a) = T4,

(iii) C◦0(g+a) \ C◦0(ga) = T5 ∪ T6,

Proof. By Theorem 3.20 and Lemma 2.16, T1 ∪ T2 ∪ T3 ∪ T4 = C◦0(ga) and T5 ∪ T6 =

C◦0(g+a) \ C◦0(ga). Let us start by proving that T1 ∪ T2 ∪ T3 ⊆ C◦0(g+

a). Suppose that G ∈ T1.

Then it is not difficult to see that G ∈ C◦0(g+a) since G+ has two blocks, one isomorphic to a

Kuratowski graph and the other consisting of a single edge.

Let G ∈ T2 and µ ∈ M(G). Since G is the xy-sum of two graphs in C◦0(g+), neither

contraction nor deletion of an edge on one part destroys the Kuratowski graph on the other

part. Thus g(µG) = 1 and M(G) ∩∆1(g) = ∅. By Lemma 2.17, G ∈ C◦0(g+a).

Let us prove now that T3 ⊆ C◦0(g+a). Consider a graph G ∈ T3. By Lemma 2.17, it is

enough to show that ∆1(g) \∆1(g+a) = ∅. Let µ ∈M(G). Let U be the set of white vertices

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 61

x

y

Figure 3.10: The graph Pinch minus a vertex. The white vertices form one part of theK3,3-subdivision.

of G as depicted in Fig. 3.8. It is not hard to show that, for each v ∈ U , G− v is nonplanar.

We omit the detailed proof of this fact and only demonstrate the proof technique on the

graph Pinch. Since U is an orbit of the isomorphism group of Pinch, it is enough to show

that G − u is nonplanar for one of the vertices u ∈ U . Indeed, G − u is isomorphic to a

subdivision of K3,3 as is exhibited in Fig. 3.10. Thus G− u is nonplanar for each u ∈ U as

required.

By Lemma 3.21, we may assume that the edge e of µ is not covered by a vertex in U .

This proves that the graphs Star, Ribbon, Five and Four are in C◦0(g+a) since U is a

vertex cover. For the other graphs, observe that the vertices in U cover all the edges not

incident with a terminal. Thus e corresponds to a label on a black vertex of G in Fig. 3.8.

Assume that µ = (e,−). By inspection, the conclusion of Lemma 3.9 is violated for G− e.Hence g+

a(µG) = 0 and µ ∈ ∆1(g+a). We may assume now that µ = (e, /). When G is one of

the graphs Saddle, Human, Alien, Bowtie, when G is Extra with e incident with the

non-terminal vertex of degree 5, and when G is Doll with e incident with the non-terminal

vertex of degree 5, the graph µG+ has the form of an xy-sum of two graphs G1 and G2.

We observe that in all cases, the graphs G+1 and G+

2 are planar and thus µG+ is planar by

Theorem 2.2. We conclude that µ ∈ ∆1(g+a). If G is Pinch, then µG is a proper minor

of Four. Since we already showed that Four ∈ C◦0(g+a), we have that µ ∈ ∆1(g+

a) in this

case as well. If G is Doll and e is incident with the black vertex of degree 3, then µG is

a proper minor of Four. The remaining case is that G is Extra and e is incident with

a non-terminal black vertex of degree 3. Again, µG is a proper minor of Five and thus

µ ∈ ∆1(g+a).

We prove that C◦0(ga) \ C◦0(g+a) = T4 by showing that T4 ∩ C◦0(g+

a) = ∅. Since each G ∈ T4

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 62

has a proper minor in T6 ⊆ C◦0(g+a), G does not belong to C◦0(g+

a). Pentagon is a minor of

Rocket and Lollipop, while Hexagon is a minor of Bullet, Frog, and Hive. Hence

T4 ⊆ C◦0(ga) \ C◦0(g+a). We have shown that the classes T1, T2, T3, T5, T6 are subclasses of

C◦0(g+a). Hence C◦0(ga) \ C◦0(g+

a) = T4.

We will use the following facts about the class C0(ga).

Lemma 3.23. For each graph G ∈ C0(ga), we have g+(G) = g(G) = 1 and hence ε(G) =

ε+(G) = θ(G) = 0.

Proof. Observe that each graph in C0(ga) is nonplanar. We shall prove that g+(G) ≤ 1

for each G ∈ T1 ∪ · · · ∪ T6 which implies that g+(G) = g(G) = 1 for each G ∈ C0(ga) by

Theorem 3.20. For a graph G ∈ T1, G+ has two blocks, one isomorphic to a Kuratowski

graph and the other consisting of a single edge. Thus g+(G) = g(G+) = 1. Each graph G in

T2 can be obtained as the xy-sum of two graphs in C◦0(g+). Theorem 2.2 gives that g(G) = 1

and θ(G) = 0 since both parts of G are xy-alternating. Hence g+(G) = 1.

To prove that a graph G ∈ T3∪T4 has g+(G) = 1, it is sufficient to provide an embedding

of G+ in the torus. Fig. 3.8 and 3.7 show that G − x − y has a drawing in the plane with

all neighbors of x and y on the outer face. Thus G/xy is a planar graph. Moreover, the

edges in the local rotation around the identified vertex in G/xy can be written as S1S2 · · ·S6

where edges in S1, S3, S5 are those incident with x in G and S2, S4, S6 are incident with y

in G. Therefore G+ admits an embedding in the torus as shown in Fig. 3.11. In the figure,

a single edge is drawn from x to the boundary of the planar patch for all the consecutive

edges that connect x and the planar patch.

We shall show that this structure of graphs in T3 ∪ T4 is not accidental. Let e ∈ E(G)

be an edge incident with x or y, say e = xv. If G − e is nonplanar, then G − e has an

xy-alternating embedding Π into the torus. The two Π-angles at x of the xy-alternating

face divide the edges in the local rotation around x into two sets, S1 and S3. Similarly, the

edges incident with y form sets S2 and S4. It is not hard to see that, since G/xy is planar,

we can pick Π so that v is Π-cofacial with y (it is not Π-cofacial with x since G is not

xy-alternating). We may assume that v lies in the region of edges in S4. Thus G/xy has the

structure described above with S5 = {e} and S4 split into sets S′4 and S6. It is thus enough

to show that there exists an edge e incident with x or y such that G− e is nonplanar. For

G ∈ T3 and an edge e ∈ E(G) incident with a white vertex in Fig. 3.8, G− e is nonplanar.

CHAPTER 3. GRAPHS CRITICAL FOR THE ALTERNATING GENUS 63

x

y

X

X

X

Y

Y

Y

Figure 3.11: An embedding of G+ in the torus for a graph G corresponding to a 3-alternatinggraph in LXY.

For G ∈ T4, the edges e such that G− e is nonplanar are depicted in Fig. 3.7 as underlined

labels.

Each graph G in T5 ∪ T6 is planar. Thus g+(G) = g(G+) ≤ 1.

We suspect that ε+(G) = ε(G) = θ(G) = 0 for all graphs in C(ga) but the proof seems

out of reach. See Chapter 3 for more details. Lemmas 2.24 and 3.23 classify when a graph

in C0(ga) ∪ C0(g+a) has θ equal to 1. We have the following corollary.

Corollary 3.24. Let G be a graph in C◦0(ga) ∪ C◦0(g+a). Then g+(G) = 1 and ε+(G) = 0.

Moreover, θ(G) = 1 if and only if G ∈ C◦0(g+a) \ C◦0(ga).

Proof. Let G ∈ C◦0(g+a) \ C◦0(ga). By Lemma 2.16, G+ ∈ C0(ga). By Lemma 2.24, θ(G) = 1

and ε+(G) = 0. Since g+a(G) = 1, g+(G) = g+

a(G)− ε+(G) = 1.

If G ∈ C◦0(ga), then G ∈ C0(ga) and thus θ(G) = ε+(G) = 0 and g+(G) = 1 by

Lemma 3.23.

Chapter 4

The Klein Bottle

The structure of obstructions for the Klein bottle is not well understood yet. Even though

no list of obstructions have been constructed so far, it is expected that the total number

of obstructions for the Klein bottle will be in tens of thousand. In this chapter, we study

g-critical graphs of low connectivity. Let Ek be the class of g-critical graphs in G with

threshold k and E =⋃k≥0 Ek. The closely related class E∗ of deletion-minimal graphs

appear naturally. Let E∗k be the class of graphs of minimum degree 3 such that g(G) > k

but g(G− e) ≤ k for each edge e ∈ E(G). Let E∗ =⋃k≥0 E∗k . The graphs in E that are not

2-connected can be obtained as disjoint unions and 1-sums of graphs in E . This is an easy

consequence of the following well-known result.

Theorem 4.1 (Stahl and Beineke [27]). The Euler genus of a graph is the sum of the Euler

genera of its blocks.

The next step is thus to study graphs in E of connectivity 2, that is, graphs that are

2-connected but not 3-connected. We shall show that each g-critical graph of connectivity

2 can be obtained as a 2-sum of two graphs that are close to graphs in E or belong to

an exceptional class of graphs, called cascades (see Section 4.2), that are studied in the

next chapter. In Section 4.3, we construct the list of g-critical graphs of connectivity 2

with threshold 2. In Section 4.4, we show that a graph of connectivity 2 is g-critical with

threshold 2 if and only if it is an obstruction for the Klein bottle. This yields a complete

list of obstructions of connectivity 2 for the Klein bottle.

64

CHAPTER 4. THE KLEIN BOTTLE 65

4.1 Euler genus of 2-sums

We can view the Euler genus of G+ as a graph parameter g+ of G, g+(G) = g(G+). Note

that g+ is minor-monotone. The difference of g+ and g is a parameter θ, that is, θ(G) =

g(G+)− g(G). Note that θ(G) ∈ {0, 1, 2}.Let G be the xy-sum of G1, G2 ∈ Gxy. We define the following two parameters:

h0(G) = g(G1) + g(G2) + 2; (4.1)

h1(G) = g+(G1) + g+(G2). (4.2)

Eq. (4.2) can be rewritten in a form similar to Eq. (4.1).

h1(G) = g(G1) + g(G2) + θ(G1) + θ(G2). (4.3)

That prompts us to define η(G1, G2) = θ(G1) + θ(G2). Note that η(G1, G2) ∈ {0, 1, 2, 3, 4}.Richter [21] gave a precise formula for the Euler genus of a 2-sum that can be expressed

using our notation as follows.

Theorem 4.2 (Richter [21]). Let G be the xy-sum of connected graphs G1, G2 ∈ Gxy. Then

(i) g(G) = min{h0(G), h1(G)},

(ii) g+(G) = h1(G), and

(iii) θ(G) = max{h1(G)− h0(G), 0}.

We can rewrite (i) as

g(G) = g(G1) + g(G2) + min{η(G1, G2), 2} (4.4)

and as

g(G) = g+(G1) + g+(G2) + 2−max{η(G1, G2), 2}. (4.5)

Since g(G) ≤ g+(G) ≤ g(G) + 2 for each G ∈ Gxy, the parameters g and g+ are 2-

separated by θ. The application of (S1) and (S2) to g and g+ gives the following (where

θ = θ(G)):

(S1) ∆k+2−θ(g) ⊆ ∆k(g

+) and

CHAPTER 4. THE KLEIN BOTTLE 66

(S2) ∆k+θ

(g+) ⊆ ∆k(g).

As an example, take a graph G with g(G) = 1 and g+(G) = 2. Then (S2) for k = 1 says

that ∆2(g+) ⊆ ∆1(g), or that each minor-operation that decreases the Euler genus of G+ by

at least 2 also decreases the Euler genus of G by at least 1.

The next lemma describes when a minor-operation in a part of a 2-sum decreases g of

the 2-sum.

Lemma 4.3. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy and let µ ∈ M(G1)

be a minor-operation such that µG1 is connected. Then g(µG) < g(G) if and only if the

following is true (where ∆k(·) always refer to the decrease of the parameter in G1):

(i) If η(G1, G2) = 0, then µ ∈ ∆1(g+).

(ii) If η(G1, G2) = 1, then µ ∈ ∆1(g+) ∪∆2(g).

(iii) If η(G1, G2) = 2, then µ ∈ ∆1(g+) ∪∆1(g).

(iv) If η(G1, G2) = 3, then µ ∈ ∆2(g+) ∪∆1(g).

(v) If η(G1, G2) = 4, then µ ∈ ∆1(g).

Proof. Assume first that g(µG) < g(G). Suppose that η(G1, G2) ≤ 2 and that µ 6∈ ∆1(g+).

Since µG1 is connected and h1(µG) = h1(G), Theorem 4.2 gives that g(µG) = h0(G). Thus

using Eq. (4.4), we obtain that

g(µG1) + g(G2) + 2 = h0(µG) = g(µG) < g(G) ≤ g(G1) + g(G2) + η(G1, G2).

If η(G1, G2) = 0, then g(µG1) < g(G1) − 2. Thus µ ∈ ∆3(g) and, since ∆3(g) ⊆ ∆1(g+)

by (S1), µ ∈ ∆1(g+), a contradiction. We conclude that (i) holds. If η(G1, G2) = 1, then

µ ∈ ∆2(g) and (ii) holds. If η(G1, G2) = 2, then µ ∈ ∆1(g) and (iii) holds.

Assume now that η(G1, G2) ≥ 3 and assume that µ 6∈ ∆1(g). Then h0(µG) = h0(G).

Consequently, by Theorem 4.2, g(µG) = h1(G). Thus using Eq. (4.5), we have

g+(µG1) + g+(G2) = h1(µG) = g(µG) < g(G) ≤ g+(G1) + g+(G2) + 2− η(G1, G2).

If η(G1, G2) = 3, then g+(µG1) < g+(G1) − 1. Hence µ ∈ ∆2(g+) and (iv) holds. If

η(G1, G2) = 4, then g+(µG1) < g+(G1)− 2. Thus µ ∈ ∆3(g+) and, since ∆3(g+) ⊆ ∆1(g) by

(S2), µ ∈ ∆1(g), a contradiction. We conclude that (v) holds.

CHAPTER 4. THE KLEIN BOTTLE 67

To prove the “if” part of the lemma, assume that (i)–(v) hold. We need to show that

g(µG) < g(G). Assume first that η(G1, G2) ≤ 2 and µ ∈ ∆1(g+). By Theorem 4.2 and (4.5),

g(µG) ≤ h0(µG) = g+(µG1) + g+(G2) < g+(G1) + g+(G2) = h0(G) = g(G).

Assume now that η(G1, G2) ≥ 2 and µ ∈ ∆1(g). By Theorem 4.2 and (4.4),

g(µG) ≤ h1(µG) = g(µG1) + g(G2) + 2 < g(G1) + g(G2) + 2 = h1(G) = g(G).

If η(G1, G2) = 1 and µ ∈ ∆2(g), then using (4.4),

g(µG) ≤ h1(µG) = g(µG1) + g(G2) + 2 < g(G1) + g(G2) + 1 = g(G).

If η(G1, G2) = 3 and µ ∈ ∆2(g+), then using (4.5),

g(µG) ≤ h0(µG) = g+(µG1) + g+(G2) < g+(G1) + g+(G2)− 1 = g(G).

Since the cases (i)–(v) cover all possible values of η, at least one of the cases above occurs

and we are done.

Let us prove a similar lemma for g+.

Lemma 4.4. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy and let µ ∈ M(G1)

be a minor-operation such that µG1 is connected. Then g+(µG) < g+(G) if and only if

µ ∈ ∆1(g+).

Proof. Assume first that g+(µG) < g+(G). Since µG1 is connected, Theorem 4.2 gives that

g+(µG) = h1(G). Using Eq. (4.2), we have that

g+(µG1) + g+(G2) = h1(µG) = g+(µG) < g+(G) = h1(G) = g+(G1) + g(G2).

Thus g+(µG1) < g+(G1). Hence µ ∈ ∆1(g+).

On the other hand, assume that µ ∈ ∆1(g+). Thus g+(µG1) < g+(G1). By Theorem 4.2

and Eq. (4.2),

g+(µG) = g+(µG1) + g+(G2) < g+(G1) + g+(G2) = g+(G),

as claimed.

CHAPTER 4. THE KLEIN BOTTLE 68

η(G1, G2) M(G1)

0 ∆1(g+)1 ∆1(g+) ∪∆2(g)2 ∆1(g+) ∪∆1(g)3 ∆2(g+) ∪∆1(g)4 ∆1(g)

Table 4.1: Possible outcomes for a minor-operation in a g-tight part of a 2-sum.

In the statements of Lemmas 4.3 and 4.4, we required that µG1 is connected. The next

lemma shows that this is indeed the case for all minor-operations if G1 is g-tight or g+-tight

in G. It is not hard to see that if G1 is a connected graph, then µG1 is connected if and

only if µ is not deletion of a cutedge of G1.

Lemma 4.5. Let G ∈ G◦xy be a connected graph with a cutedge e. Then g(G/e) = g(G) and

g+(G/e) = g+(G).

Proof. Let H1 and H2 be the components of G − e. By Theorem 4.1, g(G/e) = g(H1) +

g(H2) = g(G). If both x and y lie in H1 (or H2 by symmetry), then by Theorem 4.1,

g+(G/e) = g(H1 + xy) + g(H2) = g+(G). Suppose then that x ∈ V (H1) and y ∈ V (H2).

If x (or y by symmetry) and w are the endpoints of e, then G+ is the 1-sum of H1 and

H2 + xy + xw. Since g(H2 + yw) = g(H2 + xy + xw), we have that g+(G/e) = g(H1) +

g(H2 + yw) = g(H1) + g(H2 + xy + e) = g+(G).

Therefore we may assume that e has endpoints z ∈ V (H1) \ {x} and w ∈ V (H2) \ {y}.Let us view the graph G+ as a yz-sum of graphs H ′1 = H1 + xy and H ′2 = H2 + e. We

have that (e, /) ∈ M(H ′2) and g(H ′2/e) = g(H ′2) by Theorem 4.1 since e is a block of H ′2.

Similarly, g+(H ′2/e) = g+(H ′2) since H ′2/e is homeomorphic to H ′2 and thus admits the same

embeddings. By applying Theorem 4.1 to G+ as a yz-sum of H ′1 and H ′2, we obtain that

g(G+/e) = g(G+). We conclude that g+(G/e) = g+(G).

Lemma 4.5 easily implies that a g-tight or g+-tight part G1 of an xy-sum G has no

cutedges. If e is a cutedge of G1, then G1/e is connected, g(G1/e) = g(G1), and g+(G1/e) =

g+(G1). By Lemmas 4.3 and 4.4), G1 is not g-tight nor g+-tight in G. In particular, we may

present the outcome of Lemma 4.3 in terms of M(G1) as in Table 4.1.

CHAPTER 4. THE KLEIN BOTTLE 69

4.2 Critical classes, cascades, and hoppers

In this section, we describe the relation between the classes C◦(g), C◦(g+), and E . The next

result follows from the definitions of E and C◦(g).

Lemma 4.6. For G ∈ G◦xy, G ∈ E if and only if G ∈ C◦(g).

The next two lemmas describe the relation between the class C◦(g+) and E .

Lemma 4.7. For G ∈ G◦xy, G+ ∈ E if and only if G ∈ C◦(g+), θ(G) > 0, and g(G/xy) <

g+(G).

Proof. Let H = G+. Note that g(H) = g+(G) and M(H) = M(G) ∪ {(xy,−), (xy, /)}.Since g(µH) = g+(µG) for each µ ∈ M(G), we get that g(µH) < g(H) for each µ ∈ M(G)

if and only if G ∈ C◦(g+). Since H − xy ∼= G, we obtain that g(H − xy) < g(H) if and

only if θ(G) > 0. Since H/xy ∼= G/xy, we have that g(H/xy) < g(H) if and only if

g(G/xy) < g+(G). As H ∈ E if and only if g(µH) < g(H) for each µ ∈ M(H), the result

follows.

Lemma 4.8. Let G ∈ C◦(g+). If θ(G) = 0, then G ∈ E. If θ(G) > 0, then either G+∈ E,

or G+∈ E∗ and G/xy ∈ E.

Proof. If θ(G) = 0, then M(G) = ∆1(g) by (S2) and thus G ∈ C◦(g). Therefore G ∈ Eby Lemma 4.6. Suppose now that θ(G) > 0. Let H = G+. Since G ∈ C◦(g+), we have

that g(µH) < g(H) for each µ ∈ M(G). As g(H − xy) = g(G) < g(G) + θ(G) = g(H), we

have that H ∈ E∗. If g(G/xy) < g+(G), then H ∈ E (since both deletion and contraction

of xy decrease the Euler genus of H). Hence we may assume that g(G/xy) = g+(G). Let

µ ∈ M(G/xy) be a minor-operation in G/xy. Since µ is also a minor-operation in G, we

obtain that

g(µ(G/xy)) ≤ g(µG+) = g+(µG) < g+(G) = g(G/xy)

as µ(G/xy) is a minor of µG+. Since µ was chosen arbitrarily, G/xy ∈ E .

A graph G ∈ G◦xy is called a cascade if G satisfies the following properties:

(C1) M(G) = ∆1(g) ∪∆1(g+).

(C2) G 6∈ C◦(g).

CHAPTER 4. THE KLEIN BOTTLE 70

η(G1, G2) G1

0 C◦(g+)1 C◦(g+) ∪Hw(g)2 C◦(g+) ∪ C◦(g) ∪ S3 C◦(g) ∪Hw(g+)4 C◦(g)

Table 4.2: Classification of g-tight parts of a 2-sum.

(C3) G 6∈ C◦(g+).

Let S be the class of all cascades. We refine the class S according to the Euler genus.

Let Sk be the subclass of S of graphs G such that g+(G) = k+ 1. It is not hard to see that

for G ∈ Sk we have that g(G) = k.

Lemma 4.9. If G ∈ S, then θ(G) = 1.

Proof. If θ(G) = 0, then ∆1(g+) ⊆ ∆1(g) by (S2), violating (C2). If θ(G) = 2, then

∆1(g) ⊆ ∆1(g+) by (S1), violating (C3). Thus θ(G) = 1.

The following lemma is an immediate consequence of (C1)–(C3).

Lemma 4.10. Let G ∈ G◦xy. If M(G) = ∆1(g) ∪∆1(g+), then G ∈ C◦(g) ∪ C◦(g+) ∪ S.

Let G ∈ G◦xy. For a graph parameter P, a graph G is a P-hopper ifM(G) = ∆2(P). Let

H(P) be the class of P-hoppers. The subclass of H(P) of graphs with P equal to k + 1 is

denoted by Hk(P). In this chapter, we restrict our attention to g-hoppers and g+-hoppers.

Let us define two weaker forms of hoppers. We say that G is a weak g-hopper if G 6∈C◦(g+) andM(G) = ∆1(g+)∪∆2(g). Note that necessarily θ(G) = 0 by (S1); and G ∈ C◦(g)

by (S2). We say that G is a weak g+-hopper if G 6∈ C◦(g) and M(G) = ∆1(g) ∪ ∆2(g+).

Note that θ(G) = 2 by (S2) and G ∈ C◦(g+) by (S1). Let Hw(g) and Hw(g+) be the class

of weak g-hoppers and weak g+-hoppers, respectively. Let Hwk (P) be the subclass of Hw(P)

such that G ∈ Hwk (P) if P(G) = k + 1. The next result follows directly from the definition

of weak hoppers.

Lemma 4.11. Let G ∈ G◦xy. If M(G) = ∆2(g) ∪ ∆1(g+), then G ∈ C◦(g+) ∪ Hw(g). If

M(G) = ∆1(g) ∪∆2(g+), then G ∈ C◦(g) ∪Hw(g+).

Let us now present two theorems that characterize g-tight and g+-tight parts of 2-sum.

CHAPTER 4. THE KLEIN BOTTLE 71

Theorem 4.12. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy. The subgraph G1

is g-tight in G if and only if the following is true:

(i) If η(G1, G2) = 0, then G1 ∈ C◦(g+).

(ii) If η(G1, G2) = 1, then G1 ∈ C◦(g+) ∪Hw(g).

(iii) If η(G1, G2) = 2, then G1 ∈ C◦(g+) ∪ C◦(g) ∪ S.

(iv) If η(G1, G2) = 3, then G1 ∈ C◦(g) ∪Hw(g+).

(v) If η(G1, G2) = 4, then G1 ∈ C◦(g).

Proof. Assume first that G1 is g-tight. By Lemma 4.5, µG1 is connected for each µ ∈M(G1). If η(G1, G2) = 0, then M(G1) = ∆1(g+) by Lemma 4.3. Thus G1 ∈ C◦(g+).

Similarly, if η(G1, G2) = 4, then G1 ∈ C◦(g). If η(G1, G2) = 1, then M(G1) = ∆1(g+) ∪∆2(g). By Lemma 4.11, G1 ∈ C◦(g+) ∪ Hw(g). If η(G1, G2) = 3, then M(G1) = ∆1(g) ∪∆2(g+). By Lemma 4.11, G1 ∈ C◦(g) ∪ Hw(g+). Finally, if η(G1, G2) = 2, then M(G1) =

∆1(g) ∪∆1(g+). By Lemma 4.9, G1 ∈ C◦(g) ∪ C◦(g+) ∪ S.

Assume now that (i)–(v) hold. SinceM(G) = ∆1(g)∪∆1(g+) for G ∈ C◦(g)∪C◦(g+)∪S∪Hw(g)∪Hw(g+), Lemma 4.5 asserts that µG1 is connected for each µ ∈M(G1). Suppose first

that G1 ∈ C◦(g). Since M(G) = ∆1(g), we obtain for each η(G1, G2) ∈ {2, 3, 4} that G1 is

g-tight by Lemma 4.3. A similar argument works if G1 ∈ C◦(g+) and η(G1, G2) ∈ {0, 1, 2}.If η(G1, G2) = 1 and G1 ∈ Hw(g), then M(G1) = ∆1(g+) ∪ ∆2(g) and G1 is g-tight by

Lemma 4.3. If η(G1, G2) = 3 and G1 ∈ Hw(g+), then M(G1) = ∆2(g+) ∪∆1(g) and G1 is

g-tight by Lemma 4.3. If η(G1, G2) = 2 and G1 ∈ S, thenM(G1) = ∆1(g)∪∆1(g+) by (C1)

and G1 is g-tight by Lemma 4.3. This completes the proof since η(G1, G2) ∈ {0, . . . , 4} and

we have proven that G1 is g-tight in each case given by (i)–(v).

The outcome of Theorem 4.12 is summarized in Table 4.2. We have an analogous theorem

for g+-tight parts of 2-sums.

Theorem 4.13. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy. The subgraph G1

is g+-tight in G if and only if G1 ∈ C◦(g+).

Proof. By Lemmas 4.4 and 4.5, G1 is g+-tight if and only ifM(G1) = ∆1(g+). By definition,

G1 ∈ C◦(g+) if and only if M(G1) = ∆1(g+).

CHAPTER 4. THE KLEIN BOTTLE 72

(a)

x y

(b)

x y

(c)

x y

Figure 4.1: The class C◦0(g+), the third graph is the sole member of the class C◦0(g).

4.3 Euler genus 2

In this section, we determine the classes C◦2(g), C◦2(g+), and E2. We begin by showing that

the classes C◦0(g) and C◦0(g+) are related to Kuratowski graphs K5 and K3,3.

Lemma 4.14. The class C◦0(g) consists of a single graph, K3,3 with non-adjacent terminals

(Fig. 4.1c). The class C◦0(g+) consists of the three graphs shown in Fig. 4.1.

Proof. A graph has Euler genus greater than 0 if and only if it is non-planar. Since both K5

and K3,3 embed into projective plane, E0 = Forb(S0) = {K5,K3,3}. By Lemma 4.6, a graph

G belongs to C◦0(g) if only if G ∈ E . Since xy 6∈ E(G), G is not isomorphic to K5 and thus

C◦0(g) consists of the unique graph isomorphic to K3,3 with two non-adjacent terminals.

Let us show first that each graph in Fig. 4.1 belongs to C◦0(g+). If G+ is isomorphic to

a Kuratowski graph, then G ∈ C◦0(g+) by Lemma 4.7. Otherwise G is isomorphic to K3,3

with x and y non-adjacent. It suffices to show that µG+ is planar for each minor-operation

µ ∈M(G) as G+ clearly embeds into the projective plane. Pick an arbitrary edge e ∈ E(G).

The graph G+− e has 9 edges and is not isomorphic to K3,3 as it contains a triangle. The

graph G+/e has only 5 vertices and (at most) 9 edges. Since e was arbitrary, it follows that

µG+ is planar for every µ ∈M(G). We conclude that G ∈ C◦0(g+).

We shall show now that there are no other graphs in C◦0(g+). Let G ∈ C◦0(g+). By

Lemma 4.8, there is a graph H ∈ Forb∗(S0) such that either G is isomorphic to H or

G is isomorphic to the graph obtained from H by deleting an edge and making the ends

terminals. It is not hard to see that this yields precisely the graphs in Fig. 4.1.

Note that the first two graphs in Fig. 4.1 have θ equal to 1 and last one has θ equal to 0.

We summarize the properties of graphs in C◦0(g+) in the following lemma.

Lemma 4.15. For G ∈ C◦0(g+), G/xy is planar, θ(G) ≤ 1, and θ(G) = 1 if and only if

G 6∈ C◦0(g).

CHAPTER 4. THE KLEIN BOTTLE 73

Let us now consider the classes C◦1(g) and C◦1(g+). Since a graph embeds into the pro-

jective plane if and only if it has Euler genus at most 1, we have that E1 = Forb(N1).

Lemma 4.6 says that C◦1(g) can be constructed from the graphs G in E1 with g(G) = 2 by

choosing two nonadjacent vertices as terminals. Actually, each graph G ∈ E1 has g(G) = 2.

This construction yields 195 graphs in C◦1(g) and asserts that the list is complete. Note that

while there are 35 graphs in E1, the class C◦1(g) is larger because isomorphisms of graphs in

Gxy preserve terminals.

Lemma 4.8 provides a mean for constructing the class C◦1(g+). We construct a slightly

larger class and then test by computer which graphs are in C◦1(g+). Let G ∈ C◦1(g+). If

θ(G) = 0, then G ∈ C◦1(g) and thus G ∈ E1 ⊆ E∗1 . If θ(G) > 0, then G+ ∈ E∗1 . The class

E∗1 contains 103 graphs (see [1]). Let A be the class of graphs with terminals obtained from

E∗1 by either making two nonadjacent vertices terminals or deleting an edge e and making

the ends of e terminals. We have that, for each G ∈ C◦1(g+), G ∈ A. In order to construct

C◦1(g+), it is sufficient to check which graphs G in A are minor-minimal graphs such that

G+ does not embed into the projective plane. This construction gives 250 such graphs, out

of which only 227 graphs have G+ 2-connected. The intersection C◦1(g)∩ C◦1(g+) contains 95

graphs.

The class S1 is determined in the next chapter and shown to contain 21 graphs (and all

have G+ 2-connected). We obtained the following result using computer.

Lemma 4.16. For every G ∈ C◦1(g)∪C◦1(g+)∪S1, the graph G+ embeds into the Klein bottle.

To prove Lemma 4.16, it is sufficient to provide an embedding of G+ in the Klein bottle

for each G ∈ C◦1(g) ∪ C◦1(g+) ∪ S1. The embeddings will be available online. Based on this

evidence, we obtain the following properties of graphs in C◦1(g).

Lemma 4.17. For G ∈ C◦1(g), we have that θ(G) = 0 and ∆2(g) ⊆ ∆1(g+).

Proof. By Lemma 4.16, g+(G) = g(G+) ≤ 2. Since g(G) = 2, we have that θ(G) =

g+(G)− g(G) = 0.

The claim that ∆2(g) ⊆ ∆1(g+) was checked by computer. It is enough to show that for

each µ ∈M(G) such that µG is planar, the graph µG+ is projective planar.

Even though we do not know whether the classes Hw(g),Hw(g+),H(g), and H(g+) are

empty or not, the following subclasses are indeed empty:

CHAPTER 4. THE KLEIN BOTTLE 74

η(G1, G2) G1

0 C◦(g+)1 C◦(g+)2 C◦(g+) ∪ C◦(g) ∪ S

Table 4.3: Classification of g-tight parts of a 2-sum in C◦2(g).

Lemma 4.18. The classes Hw1 (g),Hw

1 (g+), H1(g), and H1(g+) are empty.

Proof. Let G ∈ H1(g). Since G is non-planar, it has a Kuratowski graph K as a minor.

Since g(G) = 2, K is a proper minor of G. Hence there is a minor-operation µ ∈M(G) such

that µG still has K as a minor. Thus g(µG) ≥ g(K) = 1. We conclude that µ 6∈ ∆2(g), a

contradiction.

Similarly, let G ∈ H1(g+). Since G+ is non-planar, it has a Kuratowski graph K as

a minor. Since g(G+) = 2, K is a proper minor of G+. Thus there is a minor-operation

µ ∈ M(G+) such that µG+ has K as a minor. Furthermore, since g(K + uv) = 1 for all

u, v ∈ V (G) by Lemma 4.14, we may pick µ that does not delete nor contract xy. Thus

µ ∈M(G). We have that g+(µG) ≥ g(K) = 1, a contradiction.

Let G ∈ Hw1 (g+). Thus g+(G) = 2 and θ(G) = 2. Since g(G) = 0, we have that

∆1(g) = ∅. We conclude that M(G) = ∆2(g+). Hence G ∈ H1(g+) which was already

shown to be empty.

Let G ∈ Hw1 (g). Thus g+(G) = 2, θ(G) = 0, and G ∈ C◦1(g). By Lemma 4.17, M(G) =

∆1(g+). Thus G ∈ C◦(g+), a contradiction.

Let us state some properties of the parts of xy-sums in C◦2(g) and C◦2(g+).

Lemma 4.19. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy such that g+(G1) ≤g+(G2). If G ∈ C◦2(g), then

(i) g+(G1) = 1,

(ii) g+(G2) = 2,

(iii) η(G1, G2) ≤ 2.

Proof. If g+(G2) > 2, then since G+2 is a proper subgraph of G, there is a minor-operation

µ ∈M(G) such that g(µG) ≥ g(G+2 ) > 2, a contradiction. Thus g+(G2) ≤ 2. If g+(G1) = 0,

then g(G) ≤ h1(G) = g+(G1) + g+(G2) ≤ 2 by Theorem 4.2, a contradiction. Hence

g+(G1) ≥ 1.

CHAPTER 4. THE KLEIN BOTTLE 75

By Theorem 4.2, we have

h1(G) = g+(G1) + g+(G2) ≥ g(G) = 3.

This implies that g+(G2) = 2 and (ii) holds. We also have

h0(G) = g(G1) + g(G2) + 2 ≥ g(G) = 3.

Therefore, g(G1) + g(G2) ≥ 1.

Suppose that g+(G1) = 2. If g(G1) + g(G2) ≥ 2, then by Theorem 4.2,

g(G) = min{h0(G), h1(G)} = 4,

a contradiction with g(G) = 3. Hence g(G1) + g(G2) = 1. Since g+(G1) = g+(G2), we

may exchange the roles of G1 and G2 if necessary and thus assume that g(G1) = 0. By

Lemma 4.18, H1(g+) = ∅ and thus there exists a minor-operation µ ∈ M(G1) such that

g+(µG1) ≥ 1. Note that g(µG1) = 0. By Theorem 4.2,

g(µG) = min{h0(µG), h1(µG)} = min{g(µG1) + g(G2) + 2, g+(µG1) + g+(G2)} = 3,

a contradiction with G ∈ C◦2(g). We conclude that g+(G1) = 1 and (i) holds. Since g(G1) +

g(G2) ≥ 1 and g+(G1) + g+(G2) = 3, we have that η(G1, G2) ≤ 2 and (iii) holds.

Lemma 4.20. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy such that g+(G1) ≤g+(G2). If G ∈ C◦2(g+), then

(i) g+(G1) = 1,

(ii) g+(G2) = 2.

Proof. If g+(G2) > 2, then, since G2 is a proper subgraph of G, for an arbitrary minor-

operation µ ∈ M(G1), µG still has G2 as a subgraph. Hence g+(µG) ≥ g+(G2) > 2. We

conclude that g+(G) = g+(µG) = 3, a contradiction. By Theorem 4.2,

3 = g+(G) = h1(G) = g+(G1) + g+(G2).

Since g+(G2) ≤ 2, we conclude that g+(G1) = 1 and g+(G2) = 2. Thus (i) and (ii) hold.

We are ready to state a theorem which classifies the xy-sums in C◦2(g).

CHAPTER 4. THE KLEIN BOTTLE 76

Theorem 4.21. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy. If the following

statements (i)–(iv) hold, then G ∈ C◦2(g).

(i) G1 ∈ C◦0(g+).

(ii) G2 ∈ C◦1(g+) ∪ S1.

(iii) If G1 ∈ C◦0(g), then G2 ∈ C◦1(g+).

(iv) If G1 6∈ C◦0(g), then θ(G2) ≤ 1.

Furthermore, every 2-connected graph G ∈ C◦2(g) such that {x, y} is a 2-vertex-cut can be

obtained in this way.

Proof. Let us start by proving that, if (i)–(iv) hold, then G ∈ C◦2(g). By Lemma 1.7, it is

enough to prove that G1 and G2 are g-tight in G and that g(G) = 3. If G1 ∈ C◦0(g), then

θ(G1) = 0 by Lemma 4.15. Otherwise, θ(G1) = 1 and θ(G2) ≤ 1 by (iv). We conclude

that in both cases we have η(G1, G2) ≤ 2. Theorem 4.12 and (i) give that G1 is g-tight

in G. If η(G1, G2) = 2, then G2 is g-tight in G by Theorem 4.12 and (ii). Suppose that

η(G1, G2) ≤ 1 and G2 ∈ S1. Since θ(G2) = 1 by Lemma 4.9, we have that θ(G1) = 0 and

hence G1 ∈ C◦0(g) by Lemma 4.15. This is a contradiction with (iii). Thus, we may assume

that G2 ∈ C◦1(g+). Theorem 4.12 asserts that G2 is g-tight in G. Since η(G1, G2) ≤ 2,

g+(G1) = 1, and g+(G2) = 2, Theorem 4.2 and (4.5) give that

g(G) = h1(G) = g+(G1) + g+(G2) = 3.

Therefore, G ∈ C◦2(g).

We shall now show the converse, that is, for G ∈ C◦2(g) where {x, y} is a 2-vertex-

cut, we find connected graphs G1, G2 ∈ G◦xy such that G is an xy-sum of G1 and G2

and (i)–(iv) hold. Let us distribute the {x, y}-bridges arbitrarily into G1 and G2 so that

g+(G1) ≤ g+(G2) and G1, G2 contain at least one of the bridges. By Lemma 4.19, we have

that g+(G1) = 1, g+(G2) = 2, and η(G1, G2) ≤ 2. Since Hw0 (g) and S0 are empty (see

Lemma 4.9), Theorem 4.12 gives that G1 ∈ C◦0(g) ∪ C◦0(g+) = C◦0(g+). Thus (i) holds.

Since η(G1, G2) ≤ 2, G2 ∈ C◦(g)∪C◦(g+)∪S∪Hw(g) by Theorem 4.12. By Lemma 4.18,

Hw1 (g) is empty. Since g+(G2) = 2, we have that G2 6∈ C◦0(g) ∪ C◦0(g+). We conclude that

G2 ∈ C◦1(g) ∪ C◦1(g+) ∪ S1. Assume for a contradiction that G2 ∈ C◦1(g) \ C◦1(g+). Thus there

CHAPTER 4. THE KLEIN BOTTLE 77

exists a minor-operation µ ∈ M(G2) such that µ 6∈ ∆2(g) ∪∆1(g+) since Hw1 (g) is empty.

By (4.2), h1(µG) = h1(G). Since G2 is g-tight in G, Theorem 4.2 gives:

3 = g(G) > g(µG) = h0(µG) = g(G1) + g(µG2) + 2 ≥ 3.

This contradicts our assumption that G2 6∈ C◦1(g+) ∪ S1. We conclude that (ii) holds.

Suppose that G1 ∈ C◦0(g) and G2 ∈ S1. Since θ(G1) = 0 and θ(G2) = 1 by Lemmas 4.9

and 4.15, we have that η(G1, G2) = 1. This contradicts Theorem 4.12. Thus (iii) holds.

In order to show (iv), suppose that G1 6∈ C◦0(g) and θ(G2) = 2. Then θ(G1) = 1 by

Lemma 4.15 and thus η(G1, G2) = 3. This contradicts Lemma 4.19(iii). We conclude that

(iv) holds.

We also have a corresponding theorem that classifies the xy-sums in C◦2(g+).

Theorem 4.22. Let G be the xy-sum of connected graphs G1, G2 ∈ G◦xy. If the following

statements (i) and (ii) hold, then G ∈ C◦2(g+).

(i) G1 ∈ C◦0(g+).

(ii) G2 ∈ C◦1(g+).

Furthermore, every 2-connected graph G ∈ C◦2(g+) such that {x, y} is a 2-vertex-cut can be

obtained this way.

Proof. Let us start by proving that, if (i) and (ii) hold, then G ∈ C◦2(g+). By Theorem 4.13,

G1 and G2 are g+-tight in G. Thus G ∈ C◦(g+) by Lemma 1.7. By Theorem 4.2,

g+(G) = h1(G) = g+(G1) + g+(G2) = 3.

Therefore, G ∈ C◦2(g+).

For the converse, let G1 and G2 be collections of {x, y}-bridges in G such that G is the

xy-sum of G1 and G2, g+(G1) ≤ g+(G2), and G1, G2 contain at least one of the bridges.

We shall show that (i) and (ii) hold. By Theorem 4.13, G1, G2 ∈ C◦(g+). By Lemma 4.20,

g+(G1) = 1 and g+(G2) = 2. We conclude that G1 ∈ C◦0(g+) and G2 ∈ C◦1(g+) and thus (i)

and (ii) hold.

The following lemma gives necessary and sufficient conditions for the edge xy to be

g-tight in a graph with a 2-cut {x, y} and the edge xy.

CHAPTER 4. THE KLEIN BOTTLE 78

Lemma 4.23. Let G be an xy-sum of connected graphs G1, G2 ∈ G◦xy and let H = G+. Then

the subgraph of H consisting of the edge xy is g-tight in H if and only if η(G1, G2) > 2 and

either g(G1/xy) < g+(G1) or g(G2/xy) < g+(G2),

Proof. Since g(H) = g+(G) = g(G)+ θ(G) and g(H−xy) = g(G), we have that g(H−xy) <

g(H) if and only if θ(G) > 0. Theorem 4.2 gives that θ(G) > 0 if and only if η(G1, G2) > 2.

Thus we may assume below that η(G1, G2) > 2.

By Theorem 4.1, g(H/xy) = g(G1/xy) + g(G2/xy). Since g(G1/xy) ≤ g+(G1) and

g(G2/xy) ≤ g+(G2), we have that g(H/xy) < g(H) if and only if either g(G1/xy) < g+(G1)

or g(G2/xy) < g+(G2).

We conclude this section by characterizing the graphs of connectivity 2 in E2.

Theorem 4.24. Let G be an xy-sum of connected graphs G1, G2 ∈ G◦xy such that the

following holds:

(i) G1 ∈ C◦0(g+).

(ii) G2 ∈ C◦1(g+) ∪ S1.

(iii) If G1 ∈ C◦0(g), then G2 ∈ C◦1(g+).

If η(G1, G2) ≤ 2, then G ∈ E2. If η(G1, G2) > 2, then G+∈ E2. Furthermore, each graph in

E2 of connectivity 2 is constructed this way.

Proof. Assume first that η(G1, G2) ≤ 2. By Lemma 4.6, it is sufficient to show that G1 and

G2 satisfy the conditions (i)–(iv) of Theorem 4.21. The conditions (i)–(iii) of Theorem 4.21

are the same as the assumptions of this theorem. If G1 6∈ C◦1(g), then θ(G1) = 1 by

Lemma 4.15. Since η(G1, G2) ≤ 2, we have that θ(G2) ≤ 1 and (iv) holds. By Theorem 4.21,

G ∈ C◦2(g). By Lemma 4.6, G ∈ E2.

Assume now that η(G1, G2) > 2. Since, for each graph G ∈ C◦0(g+) ∪ S1, θ(G) ≤ 1, by

Lemmas 4.9 and 4.15, we conclude that η(G1, G2) = 3, θ(G1) = 1, θ(G2) = 2, G1 6∈ C◦0(g),

and G2 ∈ C◦1(g+). By Theorem 4.22, G ∈ C◦2(g+). Note that this implies that G is g-tight

in G+. Since g+(G1/xy) < g+(G1) (Lemma 4.15), we obtain that xy is g-tight in G+ by

Lemma 4.23. Since g(G+) = g+(G) = 3, G+∈ E2 by Lemma 1.7.

Let us now prove that each H ∈ E2 of connectivity 2 is constructed this way. Pick an

arbitrary 2-vertex-cut {x, y} of H. Suppose first that xy ∈ E(H). Consider G = H − xy

CHAPTER 4. THE KLEIN BOTTLE 79

as a graph in G◦xy. Since M(G) ⊆ M(H), we have that M(G) = ∆1(g+) and G ∈ C◦(g+).

Suppose that g+(G) > 3. Let G1 and G2 be parts of G such that g+(G1) ≤ g+(G2). If

g+(G2) > 2, then for any minor-operation µ ∈ M(G1), the graph µG has G+2 as a minor.

Hence g(µH) ≥ g+(G2) > 2, a contradiction. Therefore, g+(G2) ≤ 2. By Theorem 4.2,

g+(G1) = g+(G2) = 2. Let µ ∈M(G1). By Theorem 4.2, 2 ≥ g(µH) = g+(µG) = g+(µG1)+

g+(G2). Hence g+(µG1) = 0. We conclude that M(G1) = ∆2(g+) and G1 ∈ H1(g+). By

Lemma 4.18, H1(g+) is empty, a contradiction.

So we may assume that g+(G) = 3 and thus G ∈ C◦2(g+). By Theorem 4.22, G is an

xy-sum of graphs G1 ∈ C◦0(g+) and G2 ∈ C◦1(g+). By Lemma 4.23, η(G1, G2) > 2. Thus G

satisfies the conditions (i)–(iii) of the theorem.

Suppose now that xy 6∈ E(H). Consider G = H as a graph in G◦xy. By Lemma 4.6,

G ∈ C◦(g). Suppose that g(G) > 3. Let G1 and G2 be parts of G such that g+(G1) ≤ g+(G2).

If g+(G2) > 2, then for any minor-operation µ ∈M(G1) so that µG1 is connected, the graph

µG has G+2 as a minor. Hence g(µH) ≥ g+(G2) > 2, a contradiction. Therefore, g+(G2) ≤ 2.

By Theorem 4.2, g+(G1) = g+(G2) = 2 and 3 < g(G) ≤ h0(G) = g(G1) + g(G2) + 2. We

may assume that g(G1) ≤ g(G2) and so g(G2) ≥ 1. Let µ ∈M(G1). By Theorem 4.2,

2 ≥ g(µH) = g(µG) = min{h0(µG), h1(µG)}.

Since h0(µG) = g(µG1) + g(G2) + 2 ≥ 3, we have that 2 ≥ h1(µG) = g+(µG1) + g+(G2).

We conclude that g+(µG1) = 0 and µ ∈ ∆2(g+). Since µ was arbitrary, G1 ∈ H1(g+). This

contradicts Lemma 4.18 which asserts that H1(g+) is empty.

Thus we may assume that g(G) = 3 and thus G ∈ C◦2(g). By Theorem 4.21, G is an

xy-sum of graphs G1 ∈ C◦0(g+) and G2 ∈ C◦1(g+) ∪ S1 and either G1 6∈ C◦0(g) or G2 6∈ S1. If

G1 ∈ C◦0(g), then θ(G1) = 0 by Lemma 4.15 and thus η(G1, G2) ≤ 2. Otherwise, θ(G1) ≤ 1

and θ(G2) ≤ 1 by Theorem 4.21(iv) and we obtain that η(G1, G2) ≤ 2. Thus G satisfies the

conditions (i)–(iii) of the theorem.

As a corollary we can construct the complete list of graphs in E2 of connectivity 2.

Corollary 4.25. There are precisely 668 g-critical graphs of connectivity 2 with threshold 2.

Proof. Let us begin by counting the number of pairs G1, G2 that satisfy the conditions (i)–

(iii) of Theorem 4.24. There are 3 graphs in C◦0(g+), there are 227 graphs G2 in C◦1(g+) such

that G+2 is 2-connected, and there are 21 graphs in S1 (for each G ∈ S1, the graph G+ is

CHAPTER 4. THE KLEIN BOTTLE 80

2-connected). That gives 744 pairs. There are only 723 pairs that satisfy the condition (iii)

of Theorem 4.24 that either G1 6∈ C◦0(g) or G2 6∈ S1.

Let G1, G2 ∈ G◦xy. There are two xy-sums that have parts isomorphic to G1 and G2 as

there are two ways how to identify two pairs of vertices. If G1 ∈ C◦0(g+), then there is an

automorphism of G1 exchanging the terminals. Hence there is only a single non-isomorphic

xy-sum G that has parts G1 ∈ C◦0(g+) and G2 ∈ C◦1(g+) ∪ S1. Since η(G1, G2) depends only

on G1 and G2, precisely one of G, G+ belongs to E2. There may be more pairs G1, G2 giving

the same graph H ∈ E2 though.

Let H ∈ E2 have connectivity 2. By Theorem 4.24, there exists an xy-sum G of connected

graphs G1 and G2 such that either G ∼= H or G+ ∼= H. Note that G+1 and G+

2 are 2-

connected. Suppose that H admits a nontrivial automorphism ψ such that ψ(V (G1)) 6=V (G1) (otherwise, it is just a combination of two automorphisms of G1 and G2). It is not

hard to see that if G+2 is 3-connected, each automorphism of H is trivial. Therefore, we

need to study graphs G2 ∈ C◦1(g+) ∪ S1 such that G+2 has connectivity 2.

There are 39 graphs G2 in C◦1(g+) such that G+2 has connectivity 2 and there are 4 graphs

G2 in S1 such that G+2 has connectivity 2 (see Fig. 5.5). It is not hard to check that the

125 pairs with G1 ∈ C◦0(g+) make only 70 non-isomorphic graphs in E2. We conclude that

there are 668 graphs of connectivity 2 in E2.

4.4 The Klein bottle

In this section, we characterize the obstructions of connectivity 2 for embedding graphs into

the Klein bottle. The nonorientable genus of G+ is a graph parameter g+ of G, g+(G) =

g(G+). Let us introduce graph parameters σ and σ+ that capture the property of being

orientably simple. Let σ = g − g and let σ+ = g+− g+. Note that σ(G) = 1 if G is

orientably simple and σ(G) = 0 otherwise.

The following lemma is an easy consequence of Lemma 1.2.

Lemma 4.26. If g+(G) is odd, then σ+(G) = 0.

Let us state the following theorem of Stahl and Beineke using our formalism.

Theorem 4.27 (Stahl and Beineke [27]). Let G = G1∪G2 be a 1-sum of G1 and G2. Then

g(G) = g(G1) + g(G2) + σ(G1)σ(G2).

CHAPTER 4. THE KLEIN BOTTLE 81

Moreover, σ(G) = σ(G1)σ(G2).

In order to describe how the nonorientable genus of a 2-sum of graphs can be computed

from the genera of its parts, let us introduce parameters h0 and h1 similar to h0 and h1.

Let G be an xy-sum of connected graphs G1, G2 ∈ Gxy. Define

h0(G) = h0(G) = g(G1) + g(G2) + 2 (4.6)

and

h1(G) = g+(G1) + g+(G2) + σ+(G1)σ+(G2). (4.7)

Let θ = g+− g. We shall use the following theorem of Richter.

Theorem 4.28 (Richter [22]). Let G be an xy-sum of connected graphs G1, G2 ∈ Gxy. Then

(i) g(G) = min{h0(G), h1(G)},

(ii) g+(G) = h1(G),

(iii) θ(G) = max{h1(G)− h0(G), 0},

(iv) σ+(G) = σ+(G1)σ+(G2), and

(v) if σ+(G) = 0 or η(G1, G2) ≥ 2, then σ(G) = 0, else σ(G) = 1.

The next lemma shows that the xy-sums of graphs with parts that are not orientably

simple are g-critical if and only if they are g-critical

Lemma 4.29. Let G be an xy-sum of connected graphs G1, G2 ∈ G◦xy, H ∈ {G, G+}, and

k ≥ 0. If σ+(G1) = σ+(G2) = 0, then H ∈ Ek if and only if H ∈ Forb(Nk).

Proof. LetH ∈ {G, G+}. By Theorem 4.28(iv) and (v), σ(G) = σ+(G) = σ+(G1)σ+(G2) = 0.

Therefore, σ(H) = 0.

Assume first that H ∈ Forb(Nk). We have that g(H) = g(H) > k. Let µ ∈ M(H).

Since g(µH) ≤ g(µH) ≤ k and µ is arbitrary, we have that H ∈ Ek.Assume now that H ∈ Ek. By Lemmas 4.6 and 4.7, G ∈ C◦(g) ∪ C◦(g+). We have that

g(H) = g(H) > k. Let µ ∈M(G1). By Lemma 4.5, µG1 is connected. By Theorem 4.28(iv)

and (v), σ(µG) = σ+(µG) = σ+(µG1)σ+(G2) = 0. Therefore, σ(µH) = 0. Hence g(µH) =

g(µH) ≤ k. Similarly g(µH) ≤ k for µ ∈ M(G2). If H = G, then H ∈ Forb(Nk).

CHAPTER 4. THE KLEIN BOTTLE 82

Assume then that H = G+. Since σ(H − xy) = σ(G) = 0, we have that g(H − xy) =

g(H − xy) ≤ k. By Theorem 4.27, σ(H/xy) = σ(G1/xy)σ(G2/xy). If σ(H/xy) = 0, then

g(H/xy) = g(H/xy) ≤ k. Otherwise, σ(G1/xy) = σ(G2/xy) = 1. Since G1/xy is a minor

of G+1 , we have that

g(G1/xy) = g(G1/xy)− σ(G1/xy) < g(G1/xy) ≤ g(G+1 ) = g+(G1) = g+(G1).

Similarly, g(G2/xy) < g+(G2). By Theorems 4.2 and 4.27,

g(H/xy) = g(G1/xy) + g(G2/xy) + σ(G1/xy)σ(G2/xy) < g+(G1) + g+(G2)

= g+(G) = g(H) = g(H) ≤ k.

We conclude that H ∈ Forb(Nk).

A corollary of Theorem 4.24 and Lemma 4.29 asserts that the class of obstructions for

the Klein bottle having connectivity 2 and the class of g-critical graphs of connectivity 2

with threshold 2 are the same.

Corollary 4.30. Let H be a graph of connectivity 2. Then H ∈ E2 if and only if H ∈Forb(N2).

Proof. Assume first that H ∈ E2. By Theorem 4.24, there is an xy-sum G of graphs

G1 ∈ C◦0(g+) and G2 ∈ C◦1(g+) ∪ S1 such that H ∈ {G, G+}. By Lemma 4.26, σ+(G1) = 0.

By Lemma 4.16, σ+(G2) = 0. By Lemma 4.29, H ∈ Forb(N2).

Assume now that H ∈ Forb(N2). Let G be an xy-sum of connected graphs G1, G2 ∈ G◦xysuch that H ∈ {G, G+}. Suppose that σ+(G1) = 1. If g+(G1) ≥ 2, then g+(G1) ≥ 3 and thus

G+1 does not embed into N2. This yields a contradiction as H has G+

1 as a proper minor.

Since g+(G1) > 0 by Lemma 1.6, we conclude that g+(G1) = 1. By Lemma 4.26, σ+(G1) =

0, a contradiction. Therefore by symmetry, σ+(G1) = σ+(G2) = 0. By Lemma 4.29,

H ∈ E2.

Chapter 5

Cascades

In this chapter, we determine the class S1, the class of cascades of Euler genus 1. In

Section 5.1 and 5.2, we develop our tools and show that each graph G ∈ S1 contains two

disjoint K-graphs. The cascades whose K-graphs are separated by a small vertex-cut are

then determined in Section 5.3. We conclude the chapter by describing how the cascades

whose K-graphs are well connected can be generated. Using computer, we construct the

class S1 as described in Section 5.4.

5.1 Separating cycles

Let C be a cycle in a graph G. For a C-bridge B in G, the B-side of C is the union of all C-

bridges of even distance from B in the overlap graph O(G,C). For a vertex v ∈ V (G)\V (C),

the v-side of C is the B-side of the C-bridge B containing v. Two vertices u, v ∈ V (G)\V (C)

are separated by C if the C-bridges containing u and v have odd distance in O(G,C). We

also say that C is (u, v)-separating .

Let G be a Π-embedded graph with the set F (Π) of Π-faces. The Π-face-distance

d∗Π(v1, v2) of v1, v2 ∈ V (G) is the number of Π-faces in the shortest sequence u0, f1,

u1, . . . , fk, uk such that u0 = v1, uk = v2, and the face fi ∈ F (Π) is incident with ui−1 and

ui, for i = 1, . . . , k. The face-distance d∗G(v1, v2) is the minimum Π-face-distance d∗Π(v1, v2)

over planar embeddings Π of G.

The following result relating number of separating cycles and the face-distance of two

vertices shall be used.

83

CHAPTER 5. CASCADES 84

Lemma 5.1 (Cabello and Mohar [7], Lemma 5.3). Let G be a planar graph and x, y ∈ V (G).

Then the maximum number of disjoint (x, y)-separating cycles in G is d∗G(x, y).

Let C be a cycle in a Π-embedded graph G and S the surface where G is 2-cell embedded

by Π. The cycle C is Π-contractible if C forms a surface-separating curve on S such that

one region of S − C is homeomorphic to an open disk (see [20] for a definition using the

combinatorial embedding).

Let P1, P2, P3 be internally disjoint paths connecting vertices u and v in G. If the cycles

P1∪P2 and P2∪P3 are Π-contractible, then the cycle P1∪P3 is also Π-contractible (see [20],

Proposition 4.3.1). This property is called 3-path-condition. Let T be a spanning tree of G.

A fundamental cycle of T is the unique cycle in T + e for an edge e ∈ E(G) \ E(T ).

Lemma 5.2. Let G be a Π-embedded graph, L a K-graph in G, and T a spanning tree of

L. Then one of the fundamental cycles of T in L is Π-noncontractible.

Proof. Suppose that all fundamental cycles of T are Π-contractible. Since fundamental

cycles of T generate the cycle space of L, the 3-path-condition gives that each cycle of L is

Π-contractible. Thus L separates the surface into three regions when L is homeomorphic to

K3,3 and into four regions when L is homeomorphic to K4. Since L is a K-graph in G, there

is a principal L-bridge B in G. But the attachments of B does not lie on a single cycle of

L and thus B cannot be embedded into any of the regions — a contradiction.

The following result is well-known, for example, see [15].

Lemma 5.3. Let G be a Π-embedded graph. If G contains two disjoint Π-noncontractible

cycles, then g(Π) ≥ 2.

The following lemma is a simple corollary of Lemmas 5.2 and 5.3.

Lemma 5.4. If G satisfies one of the following conditions, then g(G) ≥ 2.

(i) G contains two disjoint K-graphs.

(ii) G contains a Kuratowski subgraph K and a K-graph L that intersects K in at most

one half-open branch.

(iii) G contains a Kuratowski subgraph K and a K-graph L homeomorphic to K2,3 such

that K and L intersect in at most one branch P of K. Furthermore, the ends of P do

not lie on a single branch of L.

CHAPTER 5. CASCADES 85

Proof. If (i) holds, then the result follows by Lemmas 5.2 and 5.3. Suppose that (ii) holds

and that P is the branch of K with ends u and v such that V (L) ∩ V (K) ⊆ V (P ) \ {v}.The K-graph L′ in G obtained from K by deleting u is disjoint from L. The result follows

by (i).

Assume now that (iii) holds and that P is the branch of K with ends u and v. By

Lemma 5.2, since u and v do not lie on a single branch of L and L is 2-connected, we can

choose a set of fundamental cycles D in L such that no cycle in D contains both u and v (by

picking a spanning tree of L where u and v are leaves). Let C be a noncontractible cycle in

D. Since C contains at most one of u and v, say u, then K − u contains a K-graph disjoint

from C. The result follows by Lemmas 5.2 and 5.3.

5.2 Disjoint K-graphs

In this section, we shall show that for each graph in G ∈ S1, the graph G+ contains two

disjoint K-graphs. We need the following property of separating cycles.

Lemma 5.5. Let G be a planar graph, let x, y ∈ V (G) be vertices separated by a cycle

C, and let H be the x-side of C. Then there exists an (x, y)-separating cycle C ′ such that

C ′ ≤ H ∪ C and the C ′-bridges containing x and y overlap.

Proof. Pick C ′ to be an (x, y)-separating cycle in G such that C ′ ≤ H ∪ C and that the

distance of the C ′-bridge Bx containing x and the C ′-bridge By containing y in O(G,C ′) is

minimum. Let H ′ be the x-side of C ′ and note that H ′ ≤ H.

Since C ′ is (x, y)-separating, Bx and By have odd distance d in O(G,C ′). If d = 1, then

Bx and By overlap. Hence we may assume that d > 1. Let B1, B2, and B3 be the C ′-bridges

at distance 1, 2, and 3, respectively, from Bx on a shortest path from Bx to By in O(G,C ′).

Since B2 and Bx do not overlap, the cycle C ′ can be decomposed into two segments Q1 and

Q2 with ends v1 and v2 such that Q1 contains all attachments of Bx and Q2 contains all

attachments of B2. Furthermore, we can assume that v1 and v2 are attachments of B2. Let

P be a path in B2 connecting v1, v2 and let C ′′ be the cycle Q1 ∪ P . Let B be a C ′-bridge.

If B attaches to the interior of Q2, then B is a subgraph of a single C ′′-bridge B0 containing

Q2. Note that this is the case for B1 and B3 since they C ′-overlap with B2. If B does not

attach to the interior of Q2 it has the same attachments on C ′′ as on C ′. Since B1 only

attaches to Q1, we obtain that B1 overlaps with B0. It is not hard to see that B1 and the

CHAPTER 5. CASCADES 86

C ′′-bridge containing y have distance at most d − 2 in O(G,C ′′). Since C ′′ ≤ H ′ ∪ C ′, we

conclude that C ′′ ≤ H ∪ C. This contradicts the choice of C ′.

If G ∈ G◦xy, then a pre-K-graph in G is a subgraph of G homeomorphic to either K4 or

K2,3 that is a K-graph in G+. Separating cycles allow us to construct pre-K-graphs on each

side of the cycle.

Lemma 5.6. Let C be an (x, y)-separating cycle in a planar graph G ∈ G◦xy and let Bx and

By be overlapping C-bridges containing x and y, respectively. Then G contains a pre-K-

graph in C ∪Bx.

Proof. Assume first that Bx and By skew-overlap and let u1, v1 be attachments of Bx and

u2, v2 be attachments of By such that u1, u2, v1, v2 appear on C in this order. Let P be a

path connecting u1 and v1 in Bx. We see that P ∪ C is a pre-K-graph in G.

Assume now that Bx and By do not skew-overlap. Hence Bx and By have three at-

tachments u1, u2, u3 in common. Let P1, P2, P3 be three paths in Bx with one common

end u and with the other ends being u1, u2, u3, respectively. Let P be a (possibly trivial)

path connecting x and P1 ∪ P2 ∪ P3 in Bx − C and v the other end of P . If v = u, then

C ∪ P1 ∪ P2 ∪P3 is a pre-K-graph in G. If v ∈ V (P1), then let C ′ be the segment of C with

ends u2 and u3 that contains u1. We have that C ′ ∪ P1 ∪ P2 ∪ P3 is a pre-K-graph in G

homeomorphic to K2,3 with branch vertices u and u1. We construct a pre-K-graph similarly

if v ∈ V (P2) ∪ V (P3).

We have the following corollary.

Corollary 5.7. Let G be a planar graph in G◦xy. If d∗G(x, y) ≥ 2, then G contains two

disjoint pre-K-graphs.

Proof. By Lemma 5.1, there are two disjoint (x, y)-separating cycles C1 and C2 in G. Let

C1 and C2 be such that the x-side of C1 and the y-side of C2 are disjoint. By Lemma 5.5,

there is an (x, y)-separating cycle C ′1 such that the C ′1-bridges containing x and y overlap.

Similarly, there is an (x, y)-separating cycle C ′2 such that the C ′2-bridges containing x and

y overlap. Furthermore, we can pick C ′1 and C ′2 so that C ′1 is contained in the x-side of C1

and C ′2 in the y-side of C2. Therefore, C ′1 and C ′2 are disjoint. Let Bx be the C ′1-bridge

containing x and let By be the C ′2-bridge containing y. By Lemma 5.6, the graph G contains

a pre-K-graph in C ′1 ∪Bx and a pre-K-graph in C ′2 ∪By. We conclude that G contains two

disjoint pre-K-graphs.

CHAPTER 5. CASCADES 87

The following lemma relates the face-distance of x and y in a planar graph G ∈ G◦xy and

g+(G).

Lemma 5.8. Let G ∈ G◦xy be a planar graph. Then g+(G) ≥ 2 if and only if d∗G(x, y) ≥ 2.

Proof. Suppose first that there exists a planar embedding Π of G where d∗Π(x, y) ≤ 1. If

d∗Π(x, y) = 0, then G+ is planar and g+(G) = 0. Suppose then that d∗Π(x, y) = 1. Then

there exists a vertex v ∈ V (G) and two Π-faces f1, f2 incident with v such that f1 is

incident with x and f2 is incident with y. Let e1ve2 be a Π-angle of f1 and e3ve4 a Π-angle

of f2. We can write the local rotation around v as e2, S1, e3, e4, S2, e1. Let us construct

the following embedding Π′ of G+ in the projective plane. Let Π′(u) = Π(u) for each

u ∈ V (G) \ {x, y, v}. To obtain Π′(x), insert the edge xy into the local rotation Π(x) of x

between the edges e′1, e′2 where e′1, x, e

′2 is a Π-angle of f1. The local rotation Π′(y) of y is

obtained analogously. Let Π′(v) = e2, S2, e3, e1, SR2 , e4, where SR

2 is the reverse of S2. Let

Π′(e) = −1, if e ∈ {xy, e1, e4} ∪ S2, and Π′(e) = 1 otherwise. We leave it to the reader to

check that Π′ is indeed an embedding of G+ into the projective plane. Thus g+(G) ≤ 1 as

claimed.

Assume now that d∗G(x, y) ≥ 2. By Corollary 5.7, G+ contains two disjoint K-graphs.

By Lemma 5.4(i), g(G+) = g+(G) ≥ 2.

A pre-K-graph L in a planar graph G ∈ G◦xy is a z-K-graph for a terminal z ∈ {x, y} if

z ∈ V (L) and, if L is homeomorphic to K4, then z is either a branch vertex of L, and, if L is

homeomorphic to K2,3, then z lies on an open branch of L. The boundary of L is the cycle

of L that consists of all branches of L that are not incident with z. All vertices and edges

that do not lie on the boundary of L are said to be in the interior of L. A graph G ∈ G◦xycontains disjoint xy-K-graphs if it contains an x-K-graph and a y-K-graph that are disjoint.

We conclude this section by showing that each graph in S1 contains disjoint xy-K-graphs.

Lemma 5.9. Each graph in S1 contains disjoint xy-K-graphs.

Proof. Let G ∈ S1. By (C1) and (C3), there is a minor-operation µ ∈ M(G) such that µG

is planar but g+(µG) = 2. By Lemma 5.8, d∗µG(x, y) ≥ 2. By Corollary 5.7, µG contains two

disjoint pre-K-graphs. It is not hard to see that G also contains two disjoint pre-K-graphs;

denote them by Lx and Ly.

Suppose that x 6∈ Lx. By construction, there is a cycle C in Lx and a C-bridge Bx

containing x such that C ∪ Bx has Lx as a subgraph. Furthermore, there exists an edge

CHAPTER 5. CASCADES 88

xv that is not incident with Ly nor C. Consider the graph G/xv. Since Lx and Ly are

disjoint pre-K-graphs in G/xv, we have that g+(G/xv) ≥ 2. By (C1), G/xv is planar. Thus

y 6∈ V (Bx) as otherwise Lx is a K-graph in G/xv. Since Bx/xv has the same attachments

on C as Bx and G is nonplanar, we conclude that C ∪ Bx is nonplanar and thus contains

a Kuratowski subgraph K. Let e 6∈ E(C ∪Bx) be an edge of G incident with Lx such that

Ly is a pre-K-graph in G/e. Then in the graph G/e, K shares at most one vertex with

Ly. By Lemma 5.4(ii), g+(G) ≥ 2. Since G/e contains K, g(G/e) ≥ 1, a contradiction

with (C1). We conclude that x ∈ Lx. By symmetry y ∈ Ly. Therefore, G contains disjoint

xy-K-graphs.

5.3 The class S1

Let us consider a graph G ∈ S1. By Lemma 5.9, G contains an x-K-graph Lx and a y-K-

graph Ly that are disjoint. We shall assume that Lx is minimal in the sense that there is

no x-K-graph contained in Lx. Similarly take Ly minimal. Let By be the Lx-bridge that

contains Ly. Define Bx similarly. A base in G is a subgraph H of G such that H contains

Lx and Ly and they are pre-K-graphs in H. In this section, we use the structure obtained

in the previous section to construct graphs in S1 of low connectivity and find planar bases

in cascades of sufficient connectivity.

Since each graph G ∈ S1 has g+(G) = 2, G contains a graph H in C◦1(g+) as a minor.

The next lemma shows that H has to be planar.

Lemma 5.10. Let G ∈ G◦xy and let H ∈ G◦xy be a minor of G such that H ∈ C◦1(g+). If H

is nonplanar, then G 6∈ S1.

Proof. Suppose that H is a nonplanar minor of G ∈ S1. By (C3), H is a proper minor of

G. Thus, there exists a minor operation µ ∈ M(G) such that H is a minor of µG. Since

g+(H) = 2 and g+ is minor-monotone, g+(µG) ≥ g+(H) = 2. Similarly, since g(H) = 1, we

have that g(µG) ≥ g(H) = 1. Thus (C1) is violated and G 6∈ S1.

A selection of nonplanar graphs in C◦1(g+) is depicted in Fig. 5.4. The consequence of

Lemma 5.10 that a graph G ∈ S1 cannot contain a graph in Fig. 5.4 as a minor shall be used

regularly for a contradiction. Lemma 5.10 also implies that each graph G ∈ S1 contains

a planar graph in C◦1(g+) as a minor. Even though we do not use this fact here, it may

give more insight to the analysis given in the previous section. The complete list of planar

CHAPTER 5. CASCADES 89

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

Figure 5.1: The planar graphs in C◦1(g+).

CHAPTER 5. CASCADES 90

graphs in C◦1(g+) (obtained by computer) is depicted in Fig. 5.1. It is not hard to see that

Lemma 5.9 can be adapted to prove that each planar graph in C◦1(g+) contains disjoint xy-

K-graphs (without relying on verfication by computer). If Lx is homeomorphic to K4, then

either Lx contains a smaller x-K-graph or By attaches only at branch-vertices of Lx.

Lemma 5.11. If Lx is homeomorphic to K4, then the attachments of By are branch-vertices

of Lx.

Proof. Let w0 = x,w1, w2, and w3 be the branch-vertices of Lx and let Pi,j be the open

branch of Lx connecting wi and wj . Assume for a contradiction that there is an attachment

w of By on an open branch of Lx. Suppose first that w lies on P1,2. Then there is an

x-K-graph L homeomorphic to K2,3 and disjoint from Ly: The subgraph L consists of the

branch vertices w1, w2 and branches P1,3 ∪ P3,2, P1,2, and P1,0 ∪ P0,2. Since By attaches to

vertices x,w3, and w, and L is a proper subgraph of Lx, L is indeed an x-K-graph disjoint

from Ly, a contradiction to the minimality of Lx.

By symmetry, we may assume that w lies on P1,0. Let e be the edge of Lx incident

with x and P1,0. Consider the graph G′ = G/e. Since Lx/e is an x-K-graph of G disjoint

from Ly, G′+ contains two disjoint K-graphs and thus g(G′+) = 2 by Lemma 5.4(i). If

e = wx, then Lx/e is a K-graph in G′ and g(G′) ≥ 1. Otherwise, Lx/e contains a K-graph

L homeomorphic to K2,3 as follows. The branch-vertices of L are w1 and x. The branches

of L are paths P1,2 ∪P2,0, P1,3 ∪P3,0, and P1,0. Since By attaches on to vertices w2, w3, and

w, the subgraph L is a K-graph in G′ and g(G′) ≥ 1. We conclude that g(G′) = g(G) and

g(G′+) = g(G+) which violates (C1).

Since g(G) = 1, only one of Lx and Ly can be a K-graph in G.

Lemma 5.12. Either By does not attach to the interior of Lx or Bx does not attach to the

interior of Ly. Furthermore, if By attaches to the interior of Lx, then its attachment is x.

Proof. If both By and Bx attach to the interior of Lx and Ly, respectively, then we obtain,

using Lemma 5.11, that both Lx and Ly are K-graphs in G. By Lemma 5.4(i), g(G) ≥ 2, a

contradiction with G ∈ S1.

Suppose that By has an attachment in the interior of Lx that is different from x. Thus

there exists an edge e ∈ E(Lx) with both ends in the interior of Lx. Consider the graph

G/e. Since Lx/e is a K-graph in G/e, g(G/e) ≥ 1. Since Lx/e and Ly are xy-K-graphs in

G/e, g+(G/e) ≥ 2. This contradicts (C1).

CHAPTER 5. CASCADES 91

(a)

z

u1

(b)

z

u1

(c)

z

u1 u2

(d)

z

u1 u2

(e)

z

u1 u2

(f)

z

u1 u2 u3

(g)

z

u1 u2 u3

(h)

z

u1 u2 u3

(i)

z

u1 u2 u3 u4

Figure 5.2: Linkages to small sets. In each linkage, any subset of feet can be contracted.

Let k be the maximum number of pairwise disjoint paths in G connecting the boundaries

of Lx and Ly that are otherwise disjoint from Lx and Ly. Then we say that the xy-K-graphs

Lx and Ly are k-separated in G.

Let H be a planar graph that contains a subgraph L, called core, homeomorphic to K4

or K2,3 with distinguished cycle C in L and U ⊆ V (H). We will say that C is a boundary

cycle of L. The edges and vertices of L that do not lie in C are said to be in the interior of

L. We say that L is U -linked in H if there are |U | disjoint paths in H connecting C and U

that are internally disjoint from L. A U -completion of H is the graph obtained from H by

adding a new vertex adjacent to all vertices in U and to a vertex in the interior of L when

L is homeomorphic to K2,3 and to the branch-vertex of L in the interior of L when L is

homeomorphic to K4. We say that H is a U -linkage of L if L is U -linked in H and L is a

K-graph in the U -completion of H. If u ∈ U has degree at least 2 in H, then u is called a

foot of H. If u ∈ U has degree 1, then the foot of H containing u is a path from u to a first

vertex of degree at least 3. The foot containing u is also called the u-foot of H. A u-foot P

is removable if H is a (U \ {u})-linkage. The notion of a linkage will be used to describe a

pre-K-graph in G together with essential paths that attach onto it.

A set U ⊆ V (G) splits Lx and Ly in G if no U -bridge of G contains both a vertex of

Lx \ U and a vertex of Ly \ U . We say that U separates Lx from Ly in G if U ∪ {x} splits

CHAPTER 5. CASCADES 92

(a)

v0

v1v2x

v3

v4

v5

v6y

v7 v8

v9

v10

(b)

v1v2

v3

v4

v5

v6y

v8

v9

v10

(c)

v1v2

v3

v4

v5

v6y

v8

v9

v10

Figure 5.3: (a) A graph with Lx induced by x, v0, v1, v2 and Ly induced by y, v3, v4, v5, v6, v9.The set U = {v1, v2, v9} separates Ly from Lx but not Lx from Ly. (b) A U -linkage withcore Ly and feet v4v10v2, v9, v6v8v1. Each of the feet v4v10v2 and v9 is removable butv6v8v1 is not. This shows that Ly and U admit linkage (5.2g) (shown in Fig. 5.2(g)) usingF = {v3v9, v1v8, v2v10} and u1 7→ v1, u2 7→ v9, u3 7→ v2. (c) A completion of the linkage.

Lx and Ly and Lx is U -linked in G. The just-defined terms are illustrated in Fig. 5.3.

Lemma 5.13. If Lx and Ly are k-separated, then there exists a set U ⊆ V (G) of size k

such that one of the following cases occurs:

(i) U separates Lx from Ly and Ly from Lx.

(ii) U separates Lx from Ly and U ∪ {x} separates Ly from Lx.

(iii) U ∪ {y} separates Lx from Ly and U separates Ly from Lx.

Proof. By Lemma 5.12, we may assume that By does not attach to the interior of Lx. Let

P1, . . . , Pr be pairwise disjoint paths connecting Lx and Ly such that r is maximum and let

U0 be a minimum vertex-cut (of size r) that meets all paths connecting Lx and Ly. Note

that r ≥ k. Assume first that r = k. Hence U0 splits Lx and Ly. Since there are k pairwise

disjoint paths connecting the boundaries of Lx and Ly and all of them meet U0, both Lx

and Ly is U0-linked in G. We conclude that (i) holds.

Assume now that r > k. By Lemma 5.12, Bx has at most one attachment in the interior

of Ly. Thus there is only one path, say Pr, that has an end in the interior of Ly. Since

there are at most k disjoint paths joining the boundaries of Lx and Ly, we conclude that

r = k + 1. Let U1 be a minimum vertex-cut (of size k) that meets all paths connecting the

CHAPTER 5. CASCADES 93

boundaries of Lx and Ly. Thus U1 meets all the paths P1, . . . , Pk. We see that U1 ∪ {y}splits Lx and Ly. Also, the paths P1, . . . , Pr demonstrate that Lx is (U1 ∪ {y})-linked and

that Ly is U1-linked. We conclude that U1 ∪ {y} separates Lx from Ly and U1 separates Ly

from Lx. Hence (iii) holds. The case (ii) occurs in the symmetric case when By attaches to

the interior of Lx.

Let H be a U ′-linkage. Consider our cascade G. Let z ∈ {x, y} and U ⊆ V (G). We

say that Lz and U admit a U ′-linkage H if there exists a set F ⊆ E(G) such that Lz/F is

a z-K-graph in G/F and H is isomorphic to a subgraph of G/F such that U ′ is mapped

bijectively to U and the core of H is mapped to Lz/F .

Lemma 5.14. Let H be a base of G and let U ⊆ V (H), 1 ≤ |U | ≤ 4. If U separates Lx

from Ly in H, then Lx admits a linkage from Fig. 5.2 (with some of the feet possibly of

length zero).

Proof. Assume first that Lx is homeomorphic to K4. By Lemma 5.11, |U | ≤ 3. There are

three paths P1, P2, P3 connecting the branch-vertices of Lx to U . Choose the paths so that

each pair is disjoint if possible. Assume that U = {u}. By contracting the edges of P1, P2,

and P3 that are not incident with Lx, we obtain that Lx admits linkage (5.2a).

Assume now that U = {u1, u2} is of size two. Since there are two disjoint paths con-

necting Lx and U , we may assume that P1 and P2 are disjoint and connect Lx to u1 and

u2, respectively. We may also assume that P3 intersects only one of the other paths, say

P1. By contracting the edges of P1, P2, P3 that are not incident with Lx, we obtain that Lx

admits linkage (5.2c).

Assume now that U = {u1, u2, u3} is of size three. Since there are three disjoint paths

connecting Lx and U , we may assume that P1, P2, and P3 are pairwise disjoint. Thus Lx

admits linkage (5.2f).

Assume now that Lx is homeomorphic to K2,3. There are two paths P1, P2 connecting

the open branches of Lx on its boundary to U . Choose the paths so that they are disjoint

if possible. Assume that U = {u}. We see that Lx admits linkage (5.2b). Assume now that

U = {u1, u2} is of size two. Let Q1 and Q2 be disjoint paths joining the boundary of Lx

and u1 and u2, respectively. We may assume that Q1 = P1. If P2 is disjoint from P1, then

Lx admits linkage (5.2d). Otherwise we may assume that Q2 is disjoint from P2 and the

open branches of Lx. Hence Lx admits linkage (5.2e).

CHAPTER 5. CASCADES 94

Assume now that U = {u1, u2, u3} is of size three. Let Q1, Q2, Q3 be disjoint paths

joining the boundary of Lx and u1, u2, u3, respectively. We may assume that Q1 = P1. If P2

is disjoint from P1, then Lx admits linkage (5.2g). Otherwise Q2 and Q3 are disjoint from

P2 and the open branches of Lx. Thus Lx admits linkage (5.2h).

Assume now that U = {u1, u2, u3, u4} is of size four. Let Q1, Q2, Q3, Q4 be disjoint paths

joining the boundary of Lx and u1, u2, u3, u4, respectively. We may assume that Q1 = P1.

If P2 intersects first one of Q2, Q3, Q4, then Lx admits linkage (5.2i). If P2 intersects only

P1, then one of Q2, Q3, Q4 connects to an open branch of Lx which is a contradiction with

the choice of P2 as Q2, Q3, and Q4 are disjoint from P1.

The following lemma shall be used to reduce the cases when G admits linkages with

removable feet that meet up.

Lemma 5.15. If H is a base of G such that Lx admits a U1-linkage Hx and Ly admits a

U2-linkage Hy and there exists u ∈ U1∩U2 such that the u-feet of Hx and Hy are removable,

then there is a proper subbase of H. In particular, neither Lx nor Ly is a K-graph in G.

Proof. Suppose that both u-feet of Hx and Hy are removable for some u ∈ U . We may

assume by symmetry that u 6∈ V (Lx). Since Lx is U -linked, there is a nontrivial path P

connecting the boundary of Lx and u. Let uv be the edge in P incident with u. If is easy

to see that H − uv is a base of G.

Suppose for a contradiction that Lx is a K-graph in G. Consider the graph G′ = G−uv.

Since Hx − uv and Hy are a (U1 \ {u})-linkage and a (U2 \ {u})-linkage in G′, respectively,

we have that g(G′) ≥ 2. It is not hard to see that Lx is still a K-graph in G′. Thus

g(G′) ≥ 1. This yields a contradiction with (C1). The case when Ly is a K-graph in G is

done similarly.

Suppose that Lx and Ly are k-separated. By Lemma 5.13, there exists set U of size k

such that either (i), (ii), or (iii) holds. If (i) holds, then Lx and Ly are separated from each

other by U . Otherwise, we may assume that (ii) holds and Lx is separated from Ly by U

and Ly is separated from Lx by U ∪ {x}. By Lemma 5.14, Lx admit a U -linkage Hx and

Ly admits a Uy-linkage Hy from Fig. 5.2 such that Uy is either U or U ∪ {x}. Assume that

Hx and Hy are minimal (with respect to taking subgraphs). Let u1, . . . , uk be the vertices

of U to which Hx is linked as depicted in Fig. 5.2. Let u′1, . . . , u′r be the vertices of Uy in

CHAPTER 5. CASCADES 95

the order in which they are depicted in the picture of Hy is Fig. 5.2. In the following series

of lemmas we shall describe all cascades that are at most 2-separated.

Lemma 5.16. Lx and Ly are not 0-separated.

Proof. Suppose that Lx and Ly are 0-separated. Since Lx is an x-K-graph in G, there is a

path P connecting the boundary of Lx to Ly in G. Since P does not end on the boundary

of Ly, P ends on the interior of Ly. Thus Bx attaches to the interior of Ly. By symmetry,

By is attaches to the interior of Lx. This contradicts Lemma 5.12.

Lemma 5.17. If Lx and Ly are 1-separated, then G has one of the graphs in Fig. 5.5 as a

minor.

Proof. We have that Hx is one of (5.2a) or (5.2b). Assume first that Uy = U = {u1}. Thus

Hy is also one of (5.2a) or (5.2b). Let Gz be the U -bridge in G containing Lz, z ∈ {x, y}.Since U splits Lx and Ly in G, the U -bridges Gx and Gy are distinct. Since G is nonplanar,

one of Gx or Gy, say Gy by symmetry, is nonplanar by Theorem 4.1. Suppose that Gy is

not isomorphic to a Kuratowski graph. Then there exists a minor-operation µ ∈ M(Gy)

such that µGy is nonplanar. The graph µG+ contains a disjoint K-graph and Kuratowski

subgraph and thus g+(µG) ≥ 2 by Lemma 5.4(ii), a contradiction with (C1). Thus Gy is

isomorphic to either K5 or K3,3. It is not hard to see that yu ∈ E(G) in both cases. We

conclude that G has one of the graphs in Fig. 5.5 as a minor.

Assume now that Uy = U ∪ {x}. Hence Hy is one of (5.2c), (5.2d), or (5.2e). Since Ly

is linked to {u1, x}, there are two choices for the vertices u′1 and u′2. In each case, we find a

minor in G isomorphic to a nonplanar graph in C◦2(g+) which contradicts Lemma 5.10.

Case 1: Hy is (5.2c).

If u′1 = u1 and u′2 = x, then Hy contains (5.2d) as a sublinkage. Suppose then that

u′1 = x and u′2 = u1. If Hx is (5.2a), then G has (5.4a) as a minor. If Hx is (5.2b), then G

has (5.4b) as a minor.

Case 2: Hy is (5.2d).

Since Hy is symmetric, we have only a single case. If Hx is (5.2a), then G has (5.4c) as

a minor. If Hx is (5.2b), then G has (5.4d) as a minor.

Case 3: Hy is (5.2e).

CHAPTER 5. CASCADES 96

(g)

x

y

(h)

x

y

(a)

x

y

(c)

x

y

(b)

x

y

(d)

x

y

(e)

x

y

(f)

x

y

(i)

x

y

(j)

x

y

(k)

x

y

(l)

x

y

(m)

x

y

(n)

x

y

(o)

x

y

(p)

x

y

(q)

x

y

(r)

x

y

(s)

x

y

(t)

x

y

(u)

x

y

Figure 5.4: Selected nonplanar graphs in C◦1(g+).

CHAPTER 5. CASCADES 97

x

y

x

y

x

y

x

y

Figure 5.5: Cascades in S1 whose xy-K-graphs are 1-separated.

If u′1 = u1 and u′2 = x, then Hy contains (5.2d) as a sublinkage. Suppose thus that

u′1 = x and u′2 = u1. If Hx is (5.2a), then G has (5.4e) as a minor. If Hx is (5.2b), then G

has (5.4f) as a minor.

We deal with 2-separated K-graphs similarly.

Lemma 5.18. If Lx and Ly are 2-separated, then G has one of the graphs in Fig. 5.6 as a

minor.

Proof. We have that Hx is one of (5.2c), (5.2d), or (5.2e). Assume first that Uy = U =

{u1, u2}. Let Gz be the U -bridge containing Lz, z ∈ {x, y}. Since U splits Lx and Ly in

G, the U -bridges Gx and Gy are distinct. We will consider Gx and Gy as graphs in Gu1u2 ,

with terminals u1 and u2. Since G is nonplanar, Lemma 1.6 gives that either G+x or G+

y

is nonplanar. We may assume by symmetry that G+y is nonplanar. Thus Gy contains a

graph in C◦0(g+) as a minor. Suppose that there exists a minor-operation µ ∈ M(Gy) such

that µGy is nonplanar. Then µG+ contains a K-graph in Gx and a Kuratowski graph that

satisfy the conditions of Lemma 5.4(iii). By Lemma 5.4(iii), g+(µG) ≥ 2, a contradiction.

We conclude that Gy is isomorphic to one of the graphs in C◦0(g+). Since Ly is 2-linked to

u1, u2, we have that y 6∈ {u1, u2}. If Hx is (5.2c) or (5.2e), then Hx contains (5.2d) as a

sublinkage. Suppose now that Hx is (5.2d). If Gy is isomorphic to (4.1a), then G has (5.4i)

as a minor. If Gy is isomorphic to (4.1b), then G has (5.4j) as a minor. If Gy is isomorphic

to (4.1c), then G has (5.4k) or (5.4l) as a minor.

Assume now that Uy = U ∪ {x}. Hence Hy is one of (5.2f), (5.2g), or (5.2h).

Case 1: Hy is (5.2f).

This case is symmetric. If Hx is (5.2c), then G has (5.4n) as a minor. If Hx is (5.2d),

then G has (5.4m) as a minor. If Hx is (5.2e), then G has (5.4j) as a minor.

CHAPTER 5. CASCADES 98

(a)

x

y

(b)

x

y

Figure 5.6: Cascades in S1 whose xy-K-graphs are 2-separated.

Case 2: Hy is (5.2g).

Suppose that Hx is (5.2c). If u′2 = u1, then G has (5.4p) as a minor. If u′2 = u2, then G

has (5.4o) as a minor. If u′2 = x, then G has (5.6a) as a minor.

Suppose now that Hx is (5.2d). If u′2 = u1, then G has (5.4o) as a minor. If u′2 = x,

then G has (5.6b) as a minor.

Suppose now that Hx is (5.2e). Since u2-foot is removable in Hx and u′2-foot is removable

in Hy, Lemma 5.15 asserts that u2 6= u′2 (as Lx is a K-graph in G). If u′2 = u1, then G

has (5.4j) as a minor. If u′2 = x, then G has also (5.4j) as a minor.

Case 3: Hy is (5.2h).

Suppose that Hx is (5.2c). If u′1 = u1, then Hy has (5.2g) as a sublinkage. If u′1 = u2,

then G has (5.4t) as a minor. If u′1 = x, then G has (5.4r) as a minor.

Suppose that Hx is (5.2d). If u′1 = u1, then G has (5.4o) as a minor. If u′1 = x, then G

has (5.4r) as a minor.

Suppose that Hx is (5.2e). Since u2-foot is removable in Hx and u′2-foot and u′3-foot are

removable in Hy, Lemma 5.15 asserts that u′1 = u2. Then G has (5.4u) as a minor.

For xy-K-graphs that are k-separated for k ≥ 4, we shall use the fact that they admit

linkages that have many removable feet.

Lemma 5.19. Let U be a set of size k, k ≥ 4. Each U -linkage has at least k− 2 removable

feet.

Proof. Let H be a U -linkage with core L, |U | = k ≥ 4. By Lemma 5.11, L is isomorphic

to K2,3. Let P1, . . . , Pk be pairwise disjoint paths connecting L and U = {u1, . . . , uk} and

suppose that Pi ends with ui, i = 1, . . . , k. Since H is a U -linkage, there are paths Q1 and

Q2 connecting the open branches on the boundary of L to U . Suppose that Pi1 and Pi2 are

CHAPTER 5. CASCADES 99

(a)

x

y

(b)

x

y

(c)

x

y

(d)

x

y

(e)

x

y

Figure 5.7: Set B of bases of cascades in S1 whose xy-K-graphs are k-separated for k ≥ 3.

the paths Q1 and Q2 intersect closest to L. It is easy to see that, for i 6= i1, i2, the ui-foot

of H is removable. Thus H has at least k − 2 removable feet.

Let B be the class depicted in Fig. 5.7. A graph H is a planar minor of G if H is a

minor of a planar subgraph of G.

Lemma 5.20. If H is a base in G such that the xy-K-graphs in H are k-separated for

k ≥ 3, then G contains one of the graphs in B as a planar minor.

Proof. We may assume that H is a minimal base such that H is k-separated for k ≥ 3. If

k > 3, this allows us to use Lemma 5.15 to assure in general that no two u-feet of Hx and

Hy are removable.

Suppose first that k = 3. We have that Hx is one of (5.2f), (5.2g), or (5.2h). Assume

first Uy = U . Thus Hy is also one of (5.2f), (5.2g), or (5.2h).

Case 1: Hy is (5.2h).

If Hx is (5.2f), then G has (5.4h) as a minor. Suppose that Hx is (5.2g). There are two

cases by symmetry: If u′1 = u1, then G has (5.4j) as a minor. If u′1 = u2, then G has (5.4q)

as a minor.

Suppose now that Hx is (5.2h). There are two cases by symmetry: If u′1 = u1, then Hx

has (5.2g) as a sublinkage. If u′1 = u2, then G has (5.4r) as a minor. By symmetry, we may

assume now that neither Hx nor Hy is (5.2h).

Case 2: Hy is (5.2f).

If Hx is (5.2f), then G has (5.7a) as a planar minor. If Hx is (5.2g), then G has (5.7b)

as a planar minor. By symmetry, we may assume now that neither Hx nor Hy is (5.2f).

CHAPTER 5. CASCADES 100

Case 3: Hy is (5.2g).

The only case when Hx is (5.2g) remains. If u′2 = u2, then G has (5.7c) as a planar

minor. If u′2 = u1, then G has (5.7d) as a planar minor.

Assume now that Uy = U ∪ {x}. Hence Hy is (5.2i). If u′2 = x or u′4 = x, then Hy

contains linkage (5.2g) and this case was dealt with above. We may thus assume that u′1 = x.

If Hx is (5.2f), then G has (5.4g) as a minor.

Suppose now that Hx is (5.2g). By Lemma 5.15, u′2 6= u2 and u′4 6= u2. Thus u′3 = u2

and G has (5.4s) as a minor. If Hx is (5.2h), then Lemma 5.15 gives that u′2, u′4 6∈ {u2, u3, x}

which is impossible.

Suppose now that k = 4. Assume first that Lx and Ly are 4-separated and suppose that

Uy = U . Thus both Hx and Hy are (5.2i). By Lemma 5.15, {u′2, u′4} ∩ {u2, u4} = ∅. Thus

we may assume by symmetry that u′1 = u2, u′2 = u3, u′3 = u4, and u′4 = u1. We conclude

that G has (5.7e) as a planar minor.

We may assume now that Uy = U ∪ {x}. By Lemma 5.19, Hy has three removable feet.

Since Hx has two removable feet, there exists u ∈ U such that the u-feet of Hx and Hy are

removable. This contradicts Lemma 5.15.

Assume now that k > 4. By Lemma 5.19, there are at most two elements u in U such

that either u-foot of Hx or u-foot of Hy is not removable. Since |U | > 4, there exists u′ ∈ Usuch that the u′-feet of Hx and Hy are removable. By Lemma 5.15, there is a proper subbase

H ′ of H that is k-separated for k ≥ 3, a contradiction with the choice of H.

5.4 Nonplanar extensions of planar bases

Let B∗ be the class of planar graphs that contain a graph in B as a minor and that are

deletion-minimal. It is not hard to check that B∗ contains only five other graphs that are

not contained in B (see Fig. 5.8). In this section, we describe the minimal nonplanar graphs

that contains a subgraph homeomorphic to a graph in B∗. Using this description, we use

computer to determine the class S1. The graphs in S1 that have a subgraph homeomorphic

to a graph in B∗ are depicted in Fig. 5.9.

We formalize homeomorphism as follows. Let G,H be graphs. A mapping φ with domain

V (H)∪E(H) is called a homeomorphic embedding of H into G if for every two vertices v, v′

and every two edges e, e′ of H

(i) φ(v) is a vertex of G, and if v, v′ are distinct then φ(v), φ(v′) are distinct,

CHAPTER 5. CASCADES 101

(a)

x

y

(b)

x

y

(c)

x

y

(d)

x

y

(e)

x

y

Figure 5.8: The class B∗ \ B.

(ii) if e has ends v, v′, then φ(e) is a path of G with ends φ(v), φ(v′), and otherwise disjoint

from φ(V (H)), and

(iii) if e, e′ are distinct, then φ(e) and φ(e′) are edge-disjoint, and if they have a vertex in

common, then this vertex is an end of both.

We shall denote the fact that φ is a homeomorphic embedding of H into G by writing

φ : H ↪→ G. If K is a subgraph of H, then we denote by φ(K) the subgraph of G consisting

of all vertices φ(v), where v ∈ V (H), and all vertices and edges that belong to φ(e) for some

e ∈ E(K). Thus V (φ(K)) and φ(V (K)) mean different sets. It is easy to see that G has a

subgraph homeomorphic to H if and only if there is a homeomorphic embedding H ↪→ G.

An φ-bridge is simply an φ(H)-bridge; an φ-branch is an image of an edge of H.

The following result is well-known (see [20], Lemma 6.2.1).

Lemma 5.21. Let H be a 3-connected graph and φ a homeomorphic embedding of H into

a 3-connected graph G. Then there exists a homeomorphic embedding φ′ such that:

(i) φ(v) = φ(v′) for each v ∈ V (H).

(ii) φ′(e) is a path that is contained in the union of φ(e) and all local φ(e)-bridges.

(iii) There are no local φ′-bridges.

In order to apply Lemma 5.21 to a base in B∗, we need to assure that new homeomorphic

embedding still maps terminals to terminals. We will need the following lemmas.

CHAPTER 5. CASCADES 102

Lemma 5.22. Let G ∈ S1 that has a base homeomorphic to a graph in B∗. Then there is no

Kuratowski subgraph in G disjoint from Lx. Furthermore, there is no Kuratowski subgraph

K in G that is disjoint from Lx except for a single open branch of K.

Proof. Assume for a contradiction that there exists Kuratowski subgraph K in G that is

disjoint from Lx except possibly for an open branch P of K. By inspection of graphs in B∗,we may pick an edge e incident with Lx such that Lx is an x-K-graph in G/e and there is

a Kuratowski subgraph K ′ in G/e that shares at most one half-open branch with Lx. By

Lemma 5.4, g+(G/e) ≥ 2. Since G/e is nonplanar, this contradicts (C1).

Lemma 5.23. Let U be a vertex cut in G ∈ S1. If |U | ≤ 2, then each nontrivial U -bridge

in G contains either x or y.

Proof. Let B a nontrivial U -bridge that contains neither x nor y. We distinguish two cases.

Case 1: |U | = 1.

Consider the graph G1 = G−B◦. Since Kuratowski graphs are 3-connected, G1 contains

the same disjoint xy-K-graphs. Thus g+(G1) = 2 by Lemma 5.4(i). Since G1 is a proper

minor of G, g(G1) = 0 by (C1). By Theorem 4.1, B is nonplanar since G is nonplanar.

Consider an edge e ∈ E(Lx) and the graph G0 = G/e. The graph G0 is nonplanar since it

contains B as a subgraph. Also G+0 contains a Kuratowski subgraph in B and a K-graph

Ly that intersect in at most one vertex. Lemma 5.4(ii) gives that g+(G0) = 2. This is a

contradiction with (C1).

Case 2: U = {u, v} has size 2.

Consider the graph G1 = G − B◦ + uv. Since Kuratowski graphs are 3-connected, G1

contains the same disjoint xy-K-graphs. Thus g+(G1) = 2 by Lemma 5.4(i). Since G1

is a proper minor of G, g(G1) = 0 by (C1). By Lemma 1.6, B + uv is nonplanar since

G is nonplanar. We may assume by symmetry that |V (Ly) ∩ U | ≤ 1. Consider an edge

e ∈ E(Lx) \ {uv} and the graph G0 = G/e. The graph G0 is nonplanar since it contains

B + uv as a minor. Also G+0 contains a Kuratowski subgraph in B and a K-graph Ly that

intersect in at most one half-open branch. Lemma 5.4(ii) gives that g+(G0) = 2. This is a

contradiction with (C1).

Lemma 5.24. Let H be a base of G ∈ S1, and φ : H ↪→ G. If Lx is homeomorphic to K2,3

and P is the branch of Lx that contains the interior of Lx, then there are no local φ-bridges

with attachments only on P .

CHAPTER 5. CASCADES 103

Proof. Let C be the boundary of Lx which consists of the φ-branches P1, P2. Assume first

that there is an φ-bridge B0 that attaches only on the ends w1, w2 of P . By Lemma 5.23,

B0 is trivial and consists of the edge w1w2. Let G′ = G − w1w2. Since g+(G′) ≥ 2, we

have that G′ is planar by (C1). Since G is nonplanar, there are paths P3 and P4 connecting

P − w1 − w2 to P1 − w1 − w2 and P2 − w1 − w2, respectively. Let P5 be a path in By

connecting P1 −w1 −w2 and P2 −w1 −w2. It is easy to see that Lx ∪ P3 ∪ P4 ∪ P5 ∪w1w2

is a K5-minor. This contradicts Lemma 5.22.

We may assume now that all φ-bridges that attach on P attach on the interior of P . Let

B be the C-bridge that contains P . Suppose that B 6= P . Let G′ be the graph obtained

from G by replacing B with P . Since g+(G′) ≥ 2, we have that G′ is planar by (C1). Thus

B cannot be drawn inside a disk with C on the boundary. By Theorem 1.4, there three

possibilities. The option (iii) contradicts Lemma 5.22. Suppose that (i) holds and let P3, P4

be a pair of crossing paths. Since B is connected there is a path P5 connecting interiors of

P3 and P4. Thus C ∪ P3 ∪ P4 ∪ P5 is a K3,3-minor which contradicts Lemma 5.22. Suppose

now that (ii) holds and there is a tripod T in G. If T has a foot of nonzero length, then

C ∪ T has K3,3-minor. Otherwise, there is a path P5 connecting the two triads that T

consists of. Hence C ∪ T ∪ P5 is a K5-minor. In both cases, we obtain a contradiction with

Lemma 5.22.

Let H be a planar 3-connected graph and φ a homeomorphic embedding of H into G. A

well-known result of Tutte [28] says that φ(H) has a unique embedding in the plane where

each face is a cycle. Let us call each such a cycle an φ-face. An φ-path is path in G with

ends in φ(H) but otherwise disjoint from φ(H). An φ-jump is an φ-path such that no φ-face

includes both of its ends.

An φ-cross consists of two disjoint φ-paths P1, P2 with ends u1, v1 and u2, v2 (respec-

tively) on a common φ-face such that the ends appear in the order u1, u2, v1, v2. An φ-cross

P1, P2 is free if neither P1 nor P2 has its ends on φ(e) for a single e ∈ E(H) and, whenever

the ends of P1 and P2 are in V (φ(e1))∪ V (φ(e2)) for e1, e2 ∈ E(H), then e1 and e2 have no

end in common.

An φ-triad is an φ(H)-bridge B with three attachments that consists of three internally

disjoint paths P1, P2, P3 connecting the attachments to a vertex v ∈ V (G) \ V (φ(H)). Fur-

thermore, every pair of attachments of B lie on a common φ-face but no φ-face contains all

the attachments.

CHAPTER 5. CASCADES 104

Let C be an φ-face and v1, v2, v3 ∈ V (C) branch-vertices of φ(H). Let Q be a union of

at most two φ-branches with both ends in v1, v2, v3. An φ-tripod B is a subgraph of G such

that V (B) ∩ V (φ(H)) ⊆ V (Q) and B is a tripod that attaches to v1, v2, v3.

We will use the following well-known result.

Lemma 5.25. Let G be a subdivision of a 3-connected plane graph. Then each two inter-

secting faces of G share either a single branch-vertex or a single branch.

We say that a graph G ∈ G◦xy is essentially 3-connected if G+ is 3-connected. The

following lemma and its proof is adapted from [24].

Lemma 5.26. Let G ∈ S1 that has a subgraph homeomorphic to a graph H ∈ B∗. Then

there exists a homeomorphic embedding φ : H ↪→ G such that one of the following holds:

(W1) There exists an φ-jump, or

(W2) there exists a free φ-cross, or

(W3) there exists an φ-tripod, or

(W4) there exists an φ-triad.

Proof. Since H has no cutvertices, Lemma 5.23 gives that G is essentially 3-connected. By

Lemmas 5.21 and 5.24, there exists a homeomorphism φ from H into G such that there

are no local φ-bridges and terminals are mapped onto terminals by φ. Suppose that none

of (W1)–(W4) holds for φ. Let B be an φ-bridge and S the set of attachments of B. By

(W1), each two attachments elements of S lie on the same φ-face. By (W4), each triple in

S lie on the same φ-face. Let k ≥ 3 and let us assume that for each subset S′ of S of size

k, there exists an φ-face F such that S′ lie on F . We shall prove that the same holds for

each subset of S of size k + 1. Suppose for a contradiction that S0 = {v1, . . . , vk+1} is a

subset of S of size k + 1 such that there is no φ-face that contains S0. For i = 1, . . . , k + 1,

let Fi be the φ-face that contains S0 \ {vi}. Thus Fi are pairwise distinct. Since v1 and v2

belong to both F3 and F4, Lemma 5.25 gives that there is an φ-branch P12 that contains

v1 and v2. Similarly, there is an φ-branch Pij for each pair i, j = 1, . . . , k + 1. If the vertex

vi is not the end of Pij for some i and j, then the paths Pi1, . . . , Pi(k+1) are the same path

which implies that the vertices in S0 lie on a single φ-branch. This is a contradiction with

S0 not being cofacial. Thus we may assume that vi is the end of the branch Pij for each i

CHAPTER 5. CASCADES 105

and j. By Lemma 5.25, we have that k = 3. We see that the paths Pij form a subdivision

of K4. If every triple in S0 lies on a common φ-face, then G is indeed a subdivision of K4,

a contradiction with G ∈ S1. We conclude that there exists an φ-face that contains S.

Since there are no local φ-bridges, for each φ-bridge B, there exists a unique φ-face FB

such that FB contains all attachments of B. For an φ-face F , let GF be the union of all

φ-bridges whose attachments are contained in F . Since G is nonplanar, there exists an

φ-face F such that GF does not embed inside F . By (W3) and by Theorem 1.4, there is an

φ-cross P1, P2 in F . Let ui, vi be the ends of Pi, i = 1, 2. Pick P1 and P2 so that number

of pairs in {u1, v1, u2, v2} that lie on a single φ-branch is minimized. Assume first that u1

and v1 lie on a single φ-branch Q1. Since the bridge containing P1 is not local, there is a

path P3 connecting P1 and an φ-branch Q2 distinct from Q1. If also u2 and v2 lie on Q1,

then this yields a contradiction as P1 ∪ P2 ∪ P3 contains an φ-cross where the ends do not

lie on a single φ-branch. Thus we may assume that the pair u2, v2 does not share a common

φ-branch. If P3 is disjoint from P2, then P1 ∪ P2 ∪ P3 contains an φ-cross where the pairs

u1, v1 and u2, v2 do not share a common φ-branch. This again contradicts the choice of P1

and P2. If P3 intersects P2 (even only at the endpoint), then P ∪ P1 ∪ P2 ∪ P3 forms an

φ-tripod. This contradicts (W3). We may assume by symmetry that the pairs u1, v1 and

u2, v2 do not share a common φ-branch. By (W2), there are two φ-branches Q1, Q2 that

share a branch vertex and so that u1, u2 lie on Q1 and v1, v2 lie on Q2. This contradicts (W3)

as P1∪P2∪Q1∪Q2 forms an φ-tripod. We conclude that φ satisfies one of (W1)–(W4).

Even a stronger version of Lemma 5.26 can be proven.

Lemma 5.27. Let G ∈ S1 that has a subgraph homeomorphic to a graph H ∈ B∗. Then

there exists a homeomorphic embedding φ : H ↪→ G such that one of the following holds:

(T1) There exists an φ-jump, or

(T2) there exists an φ-cross that attaches onto branch-vertices of φ(H), or

(T3) there exists an φ-tripod with trivial feet that attaches onto branch-vertices of φ(H), or

(T4) there exist branch-vertices u1, u2, u3 of φ(H) such that no two of them lie on a common

φ-branch and there exists an φ-triad that attaches onto u1, u2, and u3.

Proof. Lemma 5.26 yields a homeomorphic embedding φ : H ↪→ G such that one of (W1)–

(W4) holds. Let µ ∈ M(G). If µG admits a homeomorphic embedding φ′ : H ↪→ µG

CHAPTER 5. CASCADES 106

that satisfies one of (W1)–(W4), then g+(µG) ≥ g+(H) = 2 and µG is nonplanar. This

contradicts (C1). Let us describe sufficient conditions for achievement of this contradiction.

Clearly, if µ is deletion of an edge e 6∈ φ(H) that also does not appear in the obstruction

given by (W1)–(W4), then φ,G − e contradicts (C1). If µ is a contraction of an edge

e ∈ φ(H) and one of its ends is not a terminal or a branch-vertex of φ(H) and, furthermore,

the ends of e are not attachments of the obstruction given by (W1)–(W4), then there is a

homeomorphic embedding φ′ : H ↪→ µG that satisfies one of (W1)–(W4), a contradiction.

Suppose that none of (T1)–(T4) holds. Thus one of (W2)–(W4) holds. Assume first

that (W2) or (W3) holds and let B be the union of φ-bridges as given by (W2) or (W3).

Let C be an φ-face of φ(H) that contains all attachments of B. Let us prove that C does

not intersect the interior of an xy-K-graph of φ(H). Suppose to the contrary that C is a

cycle of an x-K-graph L. Thus C ∪ B contains a K-graph of G. Let e ∈ E(φ(H)) \ E(L)

be an edge that is incident with L and not incident with C. By inspection of B∗, the graph

G/e contains two disjoint K-graphs and C ∪B contains a K-graph of G/e. This contradicts

(C1) by Lemma 5.4(i). Hence we may assume that C contains no terminals.

Assume that (W2) holds. Since (T2) does not hold, there is a free φ-cross P1, P2 such

that P1 has attachment u1 on an open branch Q of φ(H). Let u1, v1 and u2, v2 be the

attachments of P1 and P2, respectively. Let e1 and e2 be the edges of Q incident with u1.

Consider the graphs G1 = G/e1 and G2 = G/e2. Since Q is an φ-branch of length at least

2, there are homeomorphic embeddings φ1 : H ↪→ G1 and φ2 : H ↪→ G2. Suppose P1, P2

is a free φ1-cross. Then G1 is nonplanar and, since G1 has a base φ1(H), we have that

g+(G1) ≥ 2. This contradicts (C1). Thus P1, P2 is not a free φ1-cross. Similarly P1, P2 is

not a free φ2-cross. Let e1 = u1w1 and e2 = u1w2. Since P1, P2 is a free φ-cross, we may

assume that w1 6∈ {u2, v2}. Since P1, P2 is not a free φ1-cross, P1 has both ends on a single

φ1-branch Q1. If w2 ∈ {u2, v2}, then the ends of P1, P2 lie on Q∪Q1 in G, a contradiction.

Thus we may assume that w2 6∈ {u2, v2} and we obtain by symmetry that P1 has both ends

on a single φ2-branch Q2. We conclude that Q,Q1, Q2 is a subdivision of a triangle, v1 is

the common vertex of Q1 and Q2, w1 is the common vertex of Q and Q1, w2 is the common

vertex of Q and Q2, and u2, v2 lie on Q1, Q2, respectively. Let e3 be the edge of Q1 that is

incident with u2 and lie between w1 and u2. Consider the graph G3 = G/e3. Clearly, G3

satisfies either (W2) or (W3) and thus contradicts (C1).

Assume that (W3) holds. Since (T3) does not hold, either B has a nontrivial foot or B

has an attachment on an open φ-branch. Suppose first that B has a nontrivial foot P . Since

CHAPTER 5. CASCADES 107

P contains no terminals, contracting P preserves (W3). This is a contradiction with the

observation made above. Suppose now that u is an attachment of B on an open φ-branch

Q1. Let uv ∈ E(Q1) be an edge incident with u. Since u is not a terminal and v is not an

attachment of B by Lemma 1.5, G/uv contradicts (C1).

Assume that (W4) holds. By Lemma 1.5, the attachments u1, u2, u3 of B are indepen-

dent. Since (T4) does not hold, we may assume that u1 lies on an open φ-branch Q. If

u1 = x and L is the x-K-graph in φ(H), then L∪B contains a Kuratowski subgraph that is

disjoint from the y-K-graph, a contradiction with Lemma 5.22. Thus we may assume that u1

is not a terminal. Let u1w1, u1w2 be the edges incident of Q with u1 and let G1 = G/u1w1

and G2/u1w2. Since both G1 and G2 admit a homeomorphic embedding of H, they are

both planar. Thus there is an φ-face that contains the vertices w1, u2, u3 and an φ-face that

contains the vertices w2, u2, u3. It is not hard to see that u2, u3 is a 2-vertex-cut in G that

separates Lx and Ly, a contradiction.

The list of minimal graphs satisfying the conditions of Lemma 5.27 was generated by

computer and checked for graphs in S1. Assuming the implementation is correct we derive

the following theorem. We encourage other researchers to independently check our compu-

tation. A proof by hand seems to be possible but would involve detailed case analysis that

can be as error-prone as a computer program.

Theorem 5.28. The class S1 contains 21 graphs which are depicted in Fig. 5.5, 5.6, and 5.9.

Proof. Let us give detailed overview of the proof and indicate which parts of the proof rely

on computer results. Let C be the set of 21 graphs depicted in Fig. 5.5, 5.6, and 5.9. To show

that C ⊆ S1, we have to prove that each graph G ∈ C satisfies (C1)–(C3) and g+(G) = 2. We

are not aware of a faster method than computing g(µG) and g+(µG) for all minor-operations

µ ∈M(G) and then checking that (C1)–(C3) were satisfied. This was verified by computer

for the graphs in C.In order to show that S1 ⊆ C, let us consider a graph G ∈ S1. By Lemma 5.9, G contains

disjoint xy-K-graphs that are k-separated for some k ≥ 0. If k ≤ 2, then Lemmas 5.16, 5.17,

and 5.18 give that G ∈ C. If k ≥ 3, then Lemma 5.20 asserts that G has a base that is

homeomorphic to a graph H ∈ B∗. By Lemma 5.27, there is a homeomorphic embedding

of H into G such that one of (T1)–(T4) holds. By computer, we construct all those graphs

(which yields several hundred) and verify that all graphs that satisfy (C1)–(C3) belong

to C.

CHAPTER 5. CASCADES 108

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Chapter 6

Label Transitions Around a Vertex

Let G be a planar multigraph. Suppose that the edges incident with a vertex v0 ∈ V (G) are

labelled by integers 1, . . . , l. We are interested in finding an embedding of G in the plane

such that the number of label transitions around v0 is minimized. By a label transition we

mean two edges that are consecutive in the local rotation around v0 and whose labels are

different. We have seen in Chapter 3 that this is a natural problem for l = 2. We think that

the problem is interesting in general and devote this chapter to its solution. The problem

of minimizing the number of transitions may also be of interest in bioinformatics. Namely,

problems arising in genome sequencing and in relation to phylogenetic trees involve notions

very close to the minimization of transitions. Our solution in this chapter answers a question

posed by Cedric Chauve [8] in relation to a generalization of the Consecutive Ones Property

of matrices.

By deleting the vertex v0 from G and putting all edge labels onto vertices incident with

the deleted edges, we obtain an equivalent formulation of the same problem. We may assume

that the edges incident with the same vertex v 6= v0 have different labels because such edges

can always be drawn next to each other without increasing the number of label transitions.

Both representations are useful and will be treated in this chapter. Let H be the graph

obtained from G by deleting v0. For each v ∈ V (H), let λ(v) be the set of all labels of edges

joining v and v0 in G. If v is not a neighbor of v0, then λ(v) = ∅. The pair (H,λ) carries

the whole information about G and the labels of edges around v0 (assuming that the edges

incident with the same vertex v 6= v0 have different labels).

Let L be a set of labels. The graph H together with the labelling λ : V (H) → 2L is a

labelled graph. Let H be the graph obtained from a labelled graph H by adding a vertex

109

CHAPTER 6. LABEL TRANSITIONS AROUND A VERTEX 110

v0 to H and joining it to each vertex v by |λ(v)| edges and labelling these edges by elements

of λ(v). The vertex v0 is called the center of H. If the graph H is planar (which can be

checked in linear time, see [16]), we are back to an instance of the original problem.

Given (H,λ) or H, v0, and the labelling of edges incident with v0, consider an embedding

Π of H in the plane (all embeddings in this chapter are into the plane). Define the label

sequence Q = Q(Π) of Π to be the cyclic sequence of labels of edges emanating from v0

in the clockwise order of the local rotation around v0 in Π. The origin of a label L ∈ Qthat came from an edge vv0 is the vertex v. A label transition in Q is a pair of (cyclically)

consecutive labels A, B in Q such that A 6= B. The number of transitions τ(Q) of Q is the

number of label transitions in Q. The number of transitions τ(H) of H is the minimum

τ(Q(Π)) taken over all planar embeddings Π of H. When considering label transitions, the

graphs H and H are used interchangeably, i.e., we may write τ(H) instead of τ(H).

The following problem will be of our main interest:

Min-Trans. Given a planar multigraph G with edges incident to a fixed vertex v0 labelled

by 1, . . . , l and an integer k, determine if τ(G) ≤ k.

In the following, we show that Min-Trans can be solved in linear time when v0 is

nonseparating and the number of labels l is fixed.

Theorem 6.1. For every fixed integer l, there is a linear-time algorithm that determines the

minimum number of transitions τ(G) of a given planar multigraph G with edges incident to

a fixed nonseparating vertex v0 labelled by at most l different labels. In particular, the Min-

Trans problem when v0 is nonseparating is fixed-parameter tractable for the parameter l.

The proof of Theorem 6.1 is given in Sections 6.2 and 6.3. Let us observe that the

time complexity of our algorithm remains near-linear O(n1+ε) (for any ε > 0) as long as

l = o(log log log n) where n is the size of the input.

If we allow v0 to be a cutvertex, then the problem changes dramatically. Though the

formalism developed in this chapter is not sufficient to deal with this general case, we believe

that it can also be solved in linear time for every fixed l.

We also show that our algorithmic result is best possible in the sense that Min-Trans

becomes NP-complete when l is part of the input.

Theorem 6.2. Min-Trans is NP-complete if the number of labels l is unconstrained. The

problem remains NP-complete even when v0 is nonseparating and each label occurs precisely

twice.

CHAPTER 6. LABEL TRANSITIONS AROUND A VERTEX 111

The proof of Theorem 6.2 is deferred to Section 6.4.

6.1 Outline

The main ideas of our algorithm are described by the following steps. First, we simplify the

input graph G so that every 2-connected component of G − v0 is either an edge or a cycle

bounding a face. We call the resulting graph a cactus.

Having obtained the simplified graph, we consider the tree-structure of its blocks using a

dynamic programming approach. Herefrom we consider a cactus with a distinguished vertex

(root). For each pair A,B of labels, we keep information about the minimum number

of transitions ρ[A,B] assuming that the label sequence starts and ends at the root and

the sequence is prepended and appended with A and B, respectively. After establishing

Lemma 6.8 that covers concatenations of optimal label sequences, the recursive procedure

is rather straightforward, except for the case when the root has big degree. This is the hard

part and the details are presented in Section 6.3.

The main observation is that we only have to distinguish, for each block incident with

the root, the l2 values ρ[A,B] (where l is the fixed number of labels) and that the important

information is only how the values differ from ν = minA,B ρ[A,B]. The difference ρ[A,B]−νis either 0, 1, or 2. We can store the complete information about this difference in a variable

taking at most 3l2

different values. These values are called types. The blocks with the same

type need not be distinguished.

In order to determine an optimal sequence of blocks incident with the root, we transform

the problem into a problem of finding an optimum closed walk in a multigraphK on l vertices

(corresponding to the labels) in which each pair of vertices is joined by 3l2

edges (one for

each possible type). In this closed walk, the number of edges corresponding to any particular

type p must be equal to the number of blocks incident with the root whose type is equal

to p. The edge of K corresponding to the type p between vertices A and B is given weight

equal to the value of p at A,B. Although the number of closed walks is exponential, we

show that one can restrict himself to consider only a constant number of walks in K. The

details are given in Section 6.3.

Let W be an optimal closed walk in K. We consider W as a multiset of edges, the

multiplicities corresponding to the number of times each edge appears in W . The walk W

can be decomposed into two parts, a closed walk S and an eulerian multiset T of edges,

CHAPTER 6. LABEL TRANSITIONS AROUND A VERTEX 112

where S has constant size and traverses each edges traversed by W at least once. Moreover,

T contains only edges that have weight 0 and for each type p, the number np of edges in

T (counted with appropriate multiplicities of the multiset) that corresponds to p is even.

An important property of such a decomposition S, T is that there exists a multiset T ′ of

edges of K such that T ′ also satisfies the properties above and uses np edges of type p.

The set T ′ can be constructed in the following way. For each type p, we pick an edge ep

of S with weight 0 that corresponds to p. The multiset T ′ consists of the edge ep with

multiplicity np for each type p. It is not hard to check that S and T ′ form a decomposition

of a closed walk in K that has the same total weight as the original walk. Therefore, S, T ′

also decomposes an optimal closed walk. There are bounded number of closed walks S to be

considered and for each such closed walk, we define T ′ and check if the pair S, T ′ satisfies the

required conditions. This yields a constant-time algorithm with linear-time preprocessing

that determines the types of the blocks incident with the root.

6.2 Bounded Number of Labels

In this section, we develop most of the formalism needed to prove Theorem 6.1. In particular,

it is observed that we can restrict our attention to a special class of cactus graphs; also, the

basic structure of the algorithm is presented.

Let H and G be labelled graphs with labellings λ and µ, respectively. If every label

sequence of (H,λ) is also a label sequence of (G,µ), and vice versa, then H and G are said

to be equivalent .

A connected graph G is called a cactus if every block of G is either an edge or a cycle.

(A block in a connected graph G is either a cutedge or a maximal 2-connected subgraph of

G). A labelled cactus G is leaf-labelled if every endblock of G is an edge, every vertex of G

has at most one label, and a vertex of G is labelled if and only if it is a leaf.

The following lemma shows that it suffices to prove Theorem 6.1 for the case when H

is a leaf-labelled cactus. To avoid trivialities, we shall assume that there are at least two

vertices whose label set λ(v) is non-empty.

Lemma 6.3. Let H be a connected labelled graph. If H is planar, then there exists a leaf-

labelled cactus G which is equivalent to H. Furthermore, G can be constructed in linear

time.

CHAPTER 6. LABEL TRANSITIONS AROUND A VERTEX 113

Proof. We construct G from H in the following series of steps. First, we move the labels

onto leaves. After that we remove unlabelled parts of the graph and finally we remove the

insides of cycles.

Construct H ′ from H in the following manner. For each labelled vertex v ∈ V (H),

attach |λ(v)| new vertices vL, L ∈ λ(v) to v and then remove all labels from v. Label each

vL with the label L. A planar embedding of H ′ can be transformed to an embedding of

H with the same label sequence by contracting the edges vvL, v ∈ V (H), L ∈ λ(v), in H ′.

Conversely, a planar embedding of H can be transformed to an embedding of H ′ with the

same label sequence by subdividing the edges incident to the center of H. Hence H and H ′

are equivalent.

Suppose that v is a cutvertex of H ′ and a component B′ of H ′ − v contains no labels.

Since every planar embedding of H ′ − B′ = H ′ −B′ can be extended to an embedding

of H ′ with the same label sequence by embedding B′ in one of the faces around v, the

labelled graphs H ′ and H ′ −B′ are equivalent. For each cutvertex v, remove all unlabelled

components of H ′ − v from H ′ to obtain H ′′ that is equivalent to H ′. It follows that every

endblock of H ′′ contains a label. The subgraph H ′′ can be easily constructed in linear time

by cutting off the appropriate endblocks of H ′.

Let Π′′ be a planar embedding of H ′′ and v0 the center of H ′′. Let G be the subgraph

of H ′′ that is formed by the vertices and edges of the facial walk in H ′′ − v0 corresponding

to the face in which v0 was embedded. Note that G contains all labelled vertices of H ′′.

We claim that G is equivalent to H ′′. Since G is a subgraph of H ′′, every embedding of H ′′

gives an embedding of G with the same label sequence. By construction, every block of G is

either an edge or a cycle. Since every endblock of H ′′ contains a label, also every endblock

of G contains a label. Thus, in every planar embedding of G, every cycle of G is a facial

cycle. Hence, an embedding of G can be extended to an embedding of H ′′ with the same

label sequence by embedding the rest of H ′′ into the facial cycles as given in Π′′. Hence G

and H ′′ are equivalent.

Since a planar embedding of H ′′ can be obtained in linear time (see for example [9]), G

can be constructed in linear time. It is not difficult to check that G is a leaf-labelled cactus

as required.

In our algorithm, we use a rooted version of graphs. A root r in a leaf-labelled cactus

H can be any vertex of H. The root is marked by a special label Lr 6∈ L. We then speak of

CHAPTER 6. LABEL TRANSITIONS AROUND A VERTEX 114

a rooted leaf-labelled cactus, or simply a cactus (H, r). The restriction on labels in a rooted

leaf-labelled cactus is slightly relaxed, every leaf still has precisely one label (possibly Lr)

and a non-leaf vertex is labelled only if it is the root. When a label sequence Q of H is cut

at the label Lr (and Lr is deleted), we obtain a linear sequence called a rooted label sequence

of H. Let Q(H) denote the set of all rooted label sequences of H. Similarly to the unrooted

graphs, two rooted graphs are equivalent if they admit the same rooted label sequences.

There exists a tree-like structure, called a PC-tree (see [25]), that captures all embeddings

of a cactus in the plane. PC-trees and their rooted version, PQ-trees, are used in testing

planarity [5], see also [9]. We note that Min-Trans (with v0 nonseparating) reduces to the

problem of minimizing the number of label transitions over all cyclic permutations of the

leaves of a PC-tree that are compatible with the PC-tree.

For every embedding Π of H, there is the flipped embedding Π′ of H where each clockwise

rotation in Π is a counter-clockwise rotation in Π′. The following lemma formulates this for

a rooted label sequence of H. For a linear sequence Q, let QR denote the sequence obtained

by reversing Q.

Lemma 6.4. Let (H, r) be a rooted leaf-labelled cactus. If Q is a rooted label sequence of

H, then the reversed sequence QR is also a rooted label sequence of H.

The following lemmas establish a recursive construction of rooted label sequences. Let

us recall that for a cutvertex v of H, a v-bridge in H is a subgraph of H consisting of a

connected component of H − v together with all edges joining this component and v.

Lemma 6.5. Let (H, r) be a rooted leaf-labelled cactus where r is a leaf. Let u be the

neighbor of r. If u is labelled, then H is of order 2 and has a unique rooted label sequence

Q = λ(u). Otherwise, (H, r) is equivalent to (H − r, u).

Proof. If u is labelled, then u is a leaf and H contains precisely one label λ(u) and therefore

λ(u) is the unique rooted label sequence of H. Otherwise, take an embedding of H − r in

the plane. Recall that u as the root of H − r is given a special label Lu and thus there is an

edge connecting u and the center of H − r. Subdividing this edge gives a planar embedding

of H with the same rooted label sequence. Similarly, one can obtain an embedding of H − rfrom an embedding of H with the same rooted label sequence.

Lemma 6.6. Let (H, r) be a rooted leaf-labelled cactus such that r is a cutvertex and let

B1, . . . , Bk be the r-bridges in H. Every rooted label sequence of H can be partitioned into

CHAPTER 6. LABEL TRANSITIONS AROUND A VERTEX 115

k consecutive parts where each of the k parts is a rooted label sequence of one of (Bi, r).

Conversely, if Qi is a rooted label sequence of (Bi, r) (1 ≤ i ≤ k) and (i1, . . . , ik) is a

permutation of (1, . . . , k), then the concatenation Qi1Qi2 · · ·Qik is a rooted label sequence

of H.

Proof. Suppose for a contradiction that there is a rooted label sequence Q of H with a cyclic

subsequence L1L2L3L4 (in this order) such that L1 and L3 have origins in B1 and L2, L4

have origins outside B1. Let Π be an embedding of H that corresponds to Q and v1, . . . , v4

the origins of L1, . . . , L4. Let G be the graph obtained from H by deleting the center v0

and adding an edge uv for every two consecutive edges uv0, vv0 in the local rotation around

v0. The embedding Π can be extended to a planar embedding Π′ of G such that the added

edges form a facial cycle. Since v1 and v3 are in B1, there is a path P in B1 − r joining

v1 and v3. Similarly, there is a path Q in H − (B1 − r) joining v2 and v4. Since P and Q

are disjoint and both are embedded inside C, their endvertices cannot interlace on C. This

contradiction proves the claim and implies that all labels in each Bi appear consecutively

in every rooted label sequence of H. This proves the first part of the lemma.

The second part is an easy consequence of the fact that arbitrary embeddings of Bi

(1 ≤ i ≤ k) can be combined into an embedding of H so that the cyclic order of r-bridges

around r is Bi1 , Bi2 , . . . , Bik .

Let C be a cycle of G. For v ∈ V (C), let Dv(C) be the union of v-bridges in G that do

not contain C.

Lemma 6.7. Let (H, r) be a rooted leaf-labelled cactus with r in a cycle C of length k.

If Dr(C) is empty, then every rooted label sequence Q of H can be partitioned into k − 1

(possibly empty) consecutive parts Pv, v ∈ V (C)\{r}, where Pv is a rooted label sequence of

(Dv(C), v) and Pv appear in Q in one of the two orders corresponding to the two orientations

of C.

Proof. Let Q be a rooted label sequence of H such that the conclusion of the lemma is

not true. If labels contained in one of the subgraphs Dv(C) do not form a consecutive

subsequence of Q, we obtain a contradiction as in the proof of Lemma 6.6. Suppose now

that Q contains a cyclic subsequence L1L3L2L4 of labels (in this order) such that the origins

u1, . . . , u4 of L1, . . . , L4 are inDv1(C), . . . , Dv4(C) and v1, v2, v3, v4 appear on C in this order.

Let Π be an embedding of H that corresponds to Q and let G be the graph obtained from

CHAPTER 6. LABEL TRANSITIONS AROUND A VERTEX 116

H by deleting the center v0 and adding an edge uv for every two consecutive edges uv0,

vv0 in the local rotation around v0. The embedding Π can be easily modified to a planar

embedding Π′ of G. It is easy to check that u1, . . . , u4, v1, v3 are the branch-vertices of a

subdivision of K3,3 in G, a contradiction with G being planar.

We are interested in rooted label sequences that have the minimum number of transitions.

But to combine them later on, it is important to specify the first and the last label in the

rooted label sequence. This motivates the following definition. Let Q be a set of (linear)

label sequences. We say that a sequence Q ∈ Q is AB-minimal for labels A,B ∈ L, if

τ(AQB) = min{τ(ASB) : S ∈ Q}

where AQB is the sequence obtained from Q by adding labels A and B at the beginning

and at the end of Q, respectively. A rooted label sequence Q of (H, r) is AB-minimal if

Q is AB-minimal in Q(H). Minimal sequences are composed of minimal sequences as the

following lemma shows. This allows us to restrict our attention to minimal sequences.

Lemma 6.8. Let Q be the set of all sequences that are concatenations of a sequence in Q1

and a sequence in Q2 (in this order). Then for A,B ∈ L, every AB-minimal sequence Q in

Q is a concatenation of an AC-minimal sequence in Q1 and a CB-minimal sequence in Q2

for some label C ∈ L.

Proof. Suppose that the lemma is not true for labels A,B. Let Q = Q1Q2 be an AB-

minimal sequence in Q where Qi ∈ Qi, i ∈ {1, 2}. Let C be the first label of Q2. By

assumption, either Q1 is not AC-minimal in Q1 or Q2 is not CB-minimal in Q2. If Q1 is

not AC-minimal, let Q′1 be an AC-minimal sequence in Q1. Then τ(AQ′1C) < τ(AQ1C).

It follows that

τ(AQ′1Q2B) < τ(AQ1Q2B) = τ(AQB),

a contradiction with the choice of Q. Thus, Q2 is not CB-minimal. Consequently, if Q′2 is

a CB-minimal sequence in Q2, then τ(CQ′2B) < τ(CQ2B) = τ(Q2B). Hence, we have

τ(AQ1Q′2B) ≤ τ(AQ1CQ

′2B) < τ(AQ1CQ2B) = τ(AQ1Q2B) = τ(AQB),

again a contradiction with the choice of Q.

Let (H, r) be a rooted leaf-labelled cactus. We can describe “optimal” embeddings of H

in the plane by a set of AB-minimal rooted label sequences of H, one for each pair of labels

CHAPTER 6. LABEL TRANSITIONS AROUND A VERTEX 117

A,B ∈ L. Let ρH [A,B] be the minimum number of label transitions in an AB-minimal

rooted label sequence of H. Note that the values of ρH differ by at most 2 since adding labels

A and B to a sequence increases the number of label transitions by at most 2. Hence we can

represent ρH by the minimum ρH [A,B] over all labels A,B and by the individual differences

from this minimum. Let nH be the minimum number of label transitions in a rooted label

sequence of H and let pH [A,B] = ρH [A,B] − nH . As noted above, pH [A,B] ∈ {0, 1, 2}.The function pH : L×L → {0, 1, 2} is called the type of H. It is convenient to view the type

as a number between 1 and t := 3l2, whose digits in the ternary system correspond to the

particular values pH [A,B] (for some linear ordering of all pairs (A,B) ∈ L×L). Note that

the number t of different types is a constant when the number of labels is fixed. We will

see in Lemma 6.10 that rooted cacti of the same type are interchangeable in any ordering

around a cutvertex. We call the pair (pH , nH) the descriptor of H. For simplicity, we also

call the function ρH the descriptor of H since it is easy to compute ρH from (pH , nH), and

vice versa.

Note that the “unrooted” number of transitions τ(H) can be obtained from the descriptor

of H as

τ(H) = minA∈L

ρH [A,A].

To see this, suppose first that Q is a cyclic label sequence of H (containing Lr) and A is a

label next to Lr in Q. Let R be the cyclic sequence obtained from Q by replacing Lr with

two labels A. Then R has the same number of transitions as the cyclic sequence Q − Lr.By splitting R between the two new labels A, we obtain a linear sequence ASA, where S is

a rooted label sequence of H, with the same number of transitions as in R. This shows that

τ(H) ≥ minA∈L ρH [A,A]. For the other inequality, note that from a linear sequence ASA,

where S is a rooted label sequence of H, we can obtain a cyclic label sequence of H with

smaller or equal number of label transitions by joining the ends of ASA and then deleting

the two labels A.

Next, we consider how a descriptor of a rooted leaf-labelled cactus can be computed

from descriptors of its subcacti. The first non-trivial case is when the root lies on a cycle.

Lemma 6.9. Let (H, r) be a rooted leaf-labelled cactus such that r is a vertex of degree 2

in a cycle C of length k in H. Then the descriptor of H can be computed from descriptors

of (Dv(C), v), v ∈ V (C), in time O(l3k).

Proof. In contrast with rooting at a cutvertex, the order of subgraphs Dv(C) around C

CHAPTER 6. LABEL TRANSITIONS AROUND A VERTEX 118

is fixed. Let us take an embedding of C and let r, v1, . . . , vk−1 be the vertices of C in

the clockwise order. By Lemma 6.7, every sequence in Q(H) is a concatenation of k − 1

sequences from Q(Dv1(C)), . . . ,Q(Dvk−1(C)) in this or the reverse order. Let Pi be the set

of sequences that are concatenation of k− i sequences from Q(Dvi(C)), . . . ,Q(Dvk−1)(C) in

this order. By Lemma 6.8, every AB-minimal sequence in Pi is obtained as a concatenation

of an AL-minimal sequence in Q(Dvi(C)) and an LB-minimal sequence in Pi+1 for some

L ∈ L. Let qi[A,B] be the number of transitions in an AB-minimal sequence of Pi. Since

Pk−1 = Q(Dvk−1(C)), qk−1[A,B] = ρDvk−1

(C)[A,B]. Lemma 6.8 gives that

qi[A,B] = minL∈L{ρDvi (C)[A,L] + qi+1[L,B]}, for 1 ≤ i < k − 1.

Note that q1 stores the number of transitions of all AB-minimal rooted label sequences

of H when the order of C is fixed. To allow for flipping of C, we note that an AB-minimal

rooted label sequence of H is a BA-minimal rooted sequence of H in the flipped embedding

of H. Hence

ρH [A,B] = min{q1[A,B], q1[B,A]}.

For each 1 ≤ i ≤ k − 1, we compute each of the l2 values of qi in time O(l). That gives

the overall time complexity O(l3k).

Computing the descriptor of a leaf-labelled cactus rooted at a cutvertex turns out to

be the crux. Let (H, r) be a rooted leaf-labelled cactus where r is a cutvertex of H and

let B1, . . . , Bk be the r-bridges in H. Let bH(i) be the number of r-bridges in H of type i,

i = 1, . . . , t. We view bH as an integer vector in Zt with∑t

i=1 bH(i) = k. A non-negative

integer vector b ∈ Zt is called a bridge vector and sum(b) =∑t

i=1 b(i) the sum of b. Note

that there are at most O(kt+1) different non-negative integer vectors b in Zt with the sum

at most k.

Each bridge vector b describes a problem to be solved: How to order k, k = sum(b),

bridges of types given by b around a vertex so that the number of label transitions on

the boundaries between bridges is minimized. We will see that each ordered sequence of

types in b gives a sequence of labels with each type being a connection to the next label.

Although this sequence of labels is not unique, it is a useful concept that will be heavily

used in Section 6.3. For fixed labels A,B, let RAB(n) be the set of sequences of n+ 1 labels

(L0, . . . , Ln) such that L0 = A and Ln = B. Let P = (p1, . . . , pk) be a sequence containing

all types occurring in b (with appropriate multiplicities). For such an ordering of types in

CHAPTER 6. LABEL TRANSITIONS AROUND A VERTEX 119

b and a sequence R ∈ RAB(k), R = (L0, . . . , Lk), let m(P,R) =∑k

i=1 pi[Li−1, Li]. Let

mb[A,B] be the the minimum m(P,R) taken over all orderings P of b and all sequences

R ∈ RAB(k), k = sum(b). This minimum depends only on the types of the bridges, not on

the minimum number of label transitions of the bridges.

The fact that computation of mb is a solution to the posed problem and that it gives a

way to compute the descriptor of a leaf-labelled cactus rooted at a cutvertex is made precise

in the following lemma.

Lemma 6.10. Let (H, r) be a rooted leaf-labelled cactus and let B1, . . . , Bk be the r-bridges

in H. Then

ρH [A,B] = mbH [A,B] +k∑i=1

nBi .

Proof. Let Q be an AB-minimal rooted label sequence of H. By Lemma 6.6 and repeated

application of Lemma 6.8, the sequence Q is a concatenation of sequences Qj1Qj2 · · ·Qjkwhere Qi (i = 1, . . . , k) is an Li−1Li-minimal rooted label sequence of (Bi, r) for some labels

Li such that L0 = A and Lk = B, and (j1, . . . , jk) is a permutation of (1, . . . , k). For

i = 1, . . . , k, let pi be the type of Bji . Let R = (L0, . . . , Lk) and P = (p1, . . . , pk). Then

m(P,R) =∑k

i=1 pi[Li−1, Li] and ρH [A,B] = m(P,R) +∑k

i=1 nBi .

Conversely, let P = (p1, . . . , pk) be a sequence of k types such that there is a permutation

(j1, . . . , jk) of (1, . . . , k) such that the type of Bji is pi. Let R = (L0, . . . , Lk) be a sequence of

k+1 labels and letQi be an Li−1Li-minimal rooted label sequence in (Bji , r). By Lemma 6.6,

Q = Q1, Q2, . . . , Qk is a rooted label sequence of H with τ(Q) = m(P,R) +∑k

i=1 nBi . This

completes the proof.

This gives rise to the following dynamic program. Given a non-zero bridge vector b,

there are only t possibilities for the type p of the first bridge whose label sequence starts a

minimal label sequence of H (the existence of such a bridge follows from Lemma 6.6). By

deleting the type p from b, we obtain a smaller bridge vector bp. The value mbp is computed

recursively and then combined with p to obtain mb. However, using this approach would

yield a polynomial-time algorithm that is not fixed parameter tractable (since there are

Θ(nt) bridge vectors of sum at most n). In the next section, we sidestep this problem and

present a linear-time algorithm for computing mb.

Finally, let us outline an algorithm for Min-Trans that, as we show in the next section,

yields Theorem 6.1. We assume that the input graph has at least three labels to avoid

CHAPTER 6. LABEL TRANSITIONS AROUND A VERTEX 120

trivialities.

Algorithm 1:

Input: a labelled graph G

Output: the minimum number of transitions τ(G)

1 Construct the leaf-labelled cactus H that is equivalent to G (Lemma 6.3).

2 Root H at an arbitrary unlabelled vertex r.

3 ρH ←− Descriptor(H, r).

4 Compute τ(G) from ρH .

5 return τ(G).

Function Descriptor(H, r)

Input: a rooted leaf-labelled cactus (H, r)Output: the descriptor ρH of H

1 switch according to the role of r do2 case r is a leaf and its neighbor u is labelled3 Note that F has just two vertices r and u.4 The descriptor ρH corresponds to the single-label sequence λ(u).

5 case r is a leaf and its neighbor u is not labelled6 ρH ←− Descriptor(H − r, u).

7 case r is in a cycle C and is of degree two8 foreach v ∈ V (C) \ {r} do9 Let Bv be the union of all v-bridges that do not contain C.

10 ρBv ←− Descriptor(Bv, v).

11 Use Lemma 6.9 to compute ρH from ρBv .

12 case r is a cutvertex13 foreach r-bridge Bi do14 ρBi ←− Descriptor(Bi, r).

15 Construct the bridge vector b from ρBi .16 Compute mb.17 Use Lemma 6.10 to compute ρH from mb and ρBi .

18 return ρH .

Note that throughout Algorithm 1, each vertex of H appears as a root in Descriptor

at most twice; once as a cutvertex and once either as a leaf or on a cycle. Therefore, each

CHAPTER 6. LABEL TRANSITIONS AROUND A VERTEX 121

of the cases can happen at most n times, n = |V (H)|, and the basic recursion runs in linear

time. By Lemma 6.9, the case when the root is in a cycle takes time O(l3n) since the sum of

lengths of all cycles is bounded by n. If we can compute mb for a bridge vector b in constant

time, then Algorithm 1 runs in linear time. This is the goal of the next section.

6.3 Dealing with Bridge Vectors

In the previous section, we have sketched an algorithm for computing the minimum number

of label transitions in a planar multigraph when v0 is nonseparating. In this section, we

outline an algorithm for computing mb of a bridge vector b in constant time (Lemma 6.15),

the last ingredient for the proof of Theorem 6.1. We start by observing that mb is bounded

independently of the bridge vector b. Let us recall that t = 3l2

and t is the number of types.

Lemma 6.11. Let b be a bridge vector. Then for every A,B ∈ L,

mb[A,B] ≤ 2t+ 2.

Proof. For every type p ∈ {1, . . . , t}, there are labels Ap, Bp such that p[Ap, Bp] = 0. By

Lemma 6.4, p[Bp, Ap] = 0 as well. Let k = sum(b) and let P = (p1, . . . , pk) be the sequence

of types in b in the increasing order. Let R = (L0, . . . , Lk) be the sequence of labels such

that L0 = A, Lk = B, and for i = 1, . . . , k − 1, Li = Api if i is odd and Li = Bpi if i is

even. Note that for i = 2, . . . , k − 1, if pi−1 = pi, then either pi[Li−1, Li] = pi[Api , Bpi ] or

pi[Li−1, Li] = pi[Bpi , Api ] and so pi[Li−1, Li] = 0. Since pi[A′, B′] ≤ 2 for all labels A′, B′

and there are at most t− 1 transitions between different types,

m(P,R) =k∑i=1

pi[Li−1, Li] ≤ 2(t− 1) + 4.

Thus, mb[A,B] ≤ m(P,R) ≤ 2t+ 2.

We show next that each ordering of b given by an ordering P of types in b and a label

sequence R ∈ RA,B(sum(b)) corresponds to a walk in a particular multigraph of constant

size. Let K be the complete edge-colored and edge-weighted multigraph on vertex set Lwhere two vertices A,B ∈ L are joined by t edges such that the pth edge is colored by p

and has weight p[A,B]. Note that there are t loops at every vertex of K. For a walk W in

K, the weight w(W ) of W is the sum of weights of edges in W . Let P = (p1, . . . , pk) be an

CHAPTER 6. LABEL TRANSITIONS AROUND A VERTEX 122

ordering of types in a bridge vector b and R = (L0, . . . , Lk) be a sequence of labels. The

sequences P , R generate a walk W in K of length k where in the ith step the edge Li−1Li

with color pi is used. The weight of W is m(P,R). The walk W uses b(p) edges of color

p. The converse statement also holds: A walk W that uses b(p) edges of color p gives an

ordering P of types in b and a label sequence R such that m(P,R) = w(W ). This gives the

following lemma.

Lemma 6.12. Let b be a bridge vector, A,B labels, and w an integer. There is an AB-walk

W of weight w in K such that W uses b(p) edges of color p if and only if there is an ordering

P of types in b and a label sequence R = (A, . . . , B) such that m(P,R) = w.

In the rest of this section, we will work with multisets, that is, sets where we remember

the multiplicity of the elements in the multiset. If an element is not in the multiset, we also

say that its multiplicity is 0. In the union A of two multisets A1 and A2, A = A1 ∪ A2,

the multiplicity of an element a in A is the sum of the multiplicities of a in A1 and A2.

Similarly, we define the difference of two multisets, denoted A1 − A2. A multiset A is a

submultiset of a multiset B, A ⊆ B, if, for every a ∈ A, the multiplicity of a in A is at most

the multiplicity of a in B.

For a multigraph G and a set (multiset) S of edges in G, the subgraph G(S) induced by

S is the graph with edge-set S (the set of edges in S without their multiplicity) and whose

vertices are all those vertices of G that are incident with an edge in S. A multiset S of

edges of G is Eulerian if every vertex in G is incident with an even number of edges in S

(counting multiplicities if S is a multiset). We shall use the following well known fact: A

multiset S of edges of a multigraph G are edges of some closed walk in G (that uses each

edge in S the same number of times as its multiplicity) if and only if S is Eulerian and G(S)

is connected.

If a multiset of edges S in G can be extended to an Eulerian multiset by adding some

edges from a multiset T , then the number of edges from T that has to be added to S is

bounded by |V (G)|2 as is shown in the following lemma.

Lemma 6.13. Let S, T be multisets of edges of a multigraph G on k vertices. If S can be

extended to an Eulerian multiset by adding some edges in T , then this can be done with at

most k2 edges.

Proof. Let T ′ be a subset of T such that S ∪ T ′ is Eulerian. Let T ′′ be the subset of edges

in T ′ constructed as follows. For every two vertices u and v of G such that T ′ contains an

CHAPTER 6. LABEL TRANSITIONS AROUND A VERTEX 123

odd number of edges joining u and v, put into T ′′ an arbitrary edge from T ′ connecting u

and v. Since, for a vertex v ∈ V (G), the degree of v in G(T ′′) and the degree of v in G(T ′)

have the same parity, we have that S ∪T ′′ is also Eulerian. Since there are at most k2 pairs

of vertices of G, we have that |T ′′| ≤ k2.

In the proof of Lemma 6.15, we will use the following technical lemma that says that we

can always find a small number of cycles in an Eulerian multiset S of edges of K such that

the rest of S contains an even number of edges of every color.

Lemma 6.14. Let S be an Eulerian multiset of edges in K. Let R be the set of colors p

such that there is an odd number of edges in S with color p. Then there is a collection of

k ≤ t cycles C1, . . . , Ck in K such that ∪ki=1E(C) = T ⊆ S (where the union is a union of

multisets) and there is an odd number of edges of color p in T if and only if p ∈ R.

Proof. Let C1, . . . , Cm be a cycle decomposition of S. For i = 1, . . . ,m, let xi ∈ Zt2 be the

binary vector whose pth entry counts the number of edges of color p in Ci modulo 2. Then∑mi=1 xi = x where x is the characteristic vector of R, i.e., x(p) = 1 if and only if p ∈ R.

The proof proceeds by induction on m. If m ≤ t, then we are done. Thus m > t. Since

the dimension of the vector space Zt2 is t, there are linearly dependent vectors xi1 , xi2 , . . . , xis

such that∑s

i=1 xji = 0. Remove the edges of the cycles Ci1 , . . . , Cis from S to obtain an

Eulerian multiset S′, S′ = S−∪sj=1E(Cij ), with cycle decomposition of length m− s. Since

xi1 , . . . , xis were linearly dependent, the set R′ of colors p such that there is an odd number

of edges in S′ with color p is equal to R. By the induction hypothesis, there are k cycles

C ′1, . . . , C′k in K such that ∪ki=1E(C ′i) ⊆ S′ and that have the required parity property. Since

S′ ⊆ S, the edges of these cycles form also a submultiset of S, yielding the result.

The following lemma shows that, for a bridge vector b, the value mb can be computed

in constant time.

Lemma 6.15. Let b be a bridge vector. Then for labels A,B, mb[A,B] can be computed in

time O((l2t)4l2t+1).

Proof. By Lemma 6.12, there is an AB-walk W in K that uses b(p) edges of color p and

such that w(W ) = mb[A,B]. The walk W can be viewed as a multiset of edges of K (where

the multiplicity of each edge equals the number of times the edge appears in W ) since every

AB-walk using these edges the same number of times has the same weight. We will partition

CHAPTER 6. LABEL TRANSITIONS AROUND A VERTEX 124

W into two multisets of edges, S and T , such that S contains all the “important” edges in

W and the size of S is bounded by a constant.

Let S1 be the multiset of edges in W that have positive weight. By Lemma 6.11,

|S1| ≤ w(W ) ≤ 2t + 2. Let S2 be the set of all edges of K of weight 0 that appear in

W . Note that |S2| ≤ l2t since K has l vertices and there are t loops at each vertex and t

edges joining each pair of vertices. Let S′ = S1 ∪ S2 and let e be an AB-edge in K. By

Lemma 6.13, S′ ∪ {e} can be extended to an Eulerian multiset by adding a set S3 of at

most l2 edges of W − S′. Let S′′ = S′ ∪ S3. Since S′′ ∪ {e} is connected and Eulerian,

S′′ ∪{e} is the edge set of a closed walk in G (by the fact stated before Lemma 6.13). Thus

S′′ is the multiset of edges of an AB-walk and W − S′′ is an Eulerian multiset of edges. By

Lemma 6.14, there is a multiset S4 of at most lt edges in W −S′′ such that there is an even

number of edges of each color in W − S′′ − S4. Let S = S′′ ∪ S4. Then |S| ≤ 4l2t since

|Si| ≤ l2t for each i = 1, . . . , 4. So W can be split into two multisets S and T such that

(D1) S ∪ {e} is Eulerian and has at most 4l2t edges,

(D2) K(S) is connected,

(D3) every edge in T is also present in S,

(D4) the number of edges of color p in T is even for every color p,

(D5) all edges in T have weight 0.

Let S and T be a decomposition satisfying (D1)–(D5). Let bT be the bridge vector of T .

Now, we will show that, given S and bT , we can construct a multiset T ′ with edges given by

the same bridge vector bT as T such that S and T ′ satisfy (D1)–(D5). A color p is present

in T ′ if bT (p) > 0. By (D4), bT ≡ 0 mod 2. Let G = K(S) be the graph induced by S. By

(D3) and (D5), for each color p present in T , G has an edge ep of color p and weight 0.

Take T ′ that consists of bT (p) copies of the edge ep for each color p present in T . Then

S and T ′ satisfy conditions (D1)–(D5). Since S ∪ T ′ ∪ {e} is an Eulerian multiset of edges

and K(S ∪ T ′) is connected, we obtain that S ∪ T ′ is a multiset of edges of an AB-walk W ′

in K. Since w(W ′) = w(S) = w(W ), we have that mb[A,B] ≤ w(W ′) by Lemma 6.12.

The algorithm generates all possible multisets S and then determines if there is a multiset

T satisfying (D1)–(D5) and extending S to an AB-walk. There are l2t edges in K, so there

are at most O((l2t)4l2t) choices for a multiset S of at most 4l2t edges. The conditions

CHAPTER 6. LABEL TRANSITIONS AROUND A VERTEX 125

(D1) and (D2) can be checked in O(l2t) time. The bridge vector bT is computed from

S and b in time O(t). The condition (D4) can hold only if bT ≡ 0 mod 2 and this can

be verified in time O(t). The conditions (D3) and (D5) can hold only if, for each color p

present in T , there is an edge of color p and weight 0 in K(S). This can be verified in time

O(l2t). Then there exists T such that the decomposition S, T satisfies (D1)–(D5). Thus

mb[A,B] ≤ w(S). Since there exists a decomposition S and T with w(S) = mb[A,B] that

satisfies (D1)–(D5), the exhaustive search will eventually find such a set S. Hence, the total

time is O(l2t(l2t)4l2t).

It is likely that a fixed-parameter tractable solution can also be described by the use of

min-max algebra for shortest paths, see [10] and [6], [14].

Finally, let us conclude the section by completing the proof of Theorem 6.1.

Proof of Theorem 6.1. The proof of the correctness of Algorithm 1 consists of several lem-

mas. Lemma 6.3 shows that any input graph can be transformed to an equivalent leaf-

labelled cactus. Lemmas 6.5, 6.6, and 6.7 justify our recursive approach for computing the

descriptors of rooted cacti.

The linearity of Algorithm 1 was established at the end of Section 6.2 provided that we

can compute mb in constant time. The cornerstone of the argument was that Lemma 6.10

allows us to deal with bridge vectors instead with collections of bridges. By Lemma 6.15,

we can compute mb for a bridge vector in time O(l4t(l2t)4l2t) (applying the lemma for each

pair of labels). Since there are at most n cutvertices in the graph, the algorithm runs in

time O(l4t(l2t)4l2tn).

6.4 NP-Completeness

When the number of labels is not bounded, Min-Trans becomes harder. In this section, we

give a proof of Theorem 6.2 by providing a polynomial-time reduction from the Hamiltonian

Cycle Problem (see [18]).

Ham-Cycle. For a graph G, determine if G contains a hamiltonian cycle.

Proof of Theorem 6.2. An embedding of G with at most k transitions is a certificate for

Min-Trans which asserts that Min-Trans is in NP. To show that Min-Trans is NP-

complete, we give a polynomial-time reduction from Ham-Cycle. Let G be a graph of

CHAPTER 6. LABEL TRANSITIONS AROUND A VERTEX 126

order n. Let H be the graph whose vertex set is V (H) = {w}∪V (G)∪ (E(G)×{0, 1}). We

connect w to each vertex in V (G) and for each edge uv ∈ E(G) we join one of (uv, 0) and

(uv, 1) with u, and the other one with v. Only the leaves of H are labelled. Vertex (e, i) is

labelled e. Thus, the number of labels is |E(G)| and each label occurs precisely twice. It is

immediate that H can be constructed in polynomial time in |V (G)|.We ask if the number of transitions τ(H) is smaller or equal to k for

k =∑

v∈V (G)

(deg(v)− 1) = 2|E(G)| − |V (G)|.

In the affirmative, there is a planar embedding Π of H with τ(Π) ≤ k. The local rotation

around w gives a cyclic order π of vertices of G. Root H at w. By Lemma 6.6, every

label sequence of H is a concatenation of sequences Qv, v ∈ V (G), such that Qv consists of

labels on leaves of H attached to v. Since labels in Qv are the edges adjacent to v, they are

different and thus τ(Qv) = deg(v)− 1. Hence,

τ(H) ≥∑

v∈V (G)

(deg(v)− 1) = k. (6.1)

To get an equality here, we need that there are no more label transitions between neighboring

sequences Qv.

Let e1(v) and e2(v) be the first and the last label in Qv. We have an equality in (6.1)

if and only if for every two consecutive vertices u, v in π, e1(u) = e2(v). This gives a cyclic

sequence C of n edges that visits every vertex precisely once. Hence C is a hamiltonian

cycle in G.

On the other hand, a hamiltonian cycle C in G gives a cyclic order on vertices of G.

This and the cyclic order of the edges of C give a construction of an embedding of H with

τ(H) = k.

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INDEX OF SYMBOLS 129

Index of Symbols

↪→, 101

Ak(ga), 39

B, 99B∗, 100B◦, 2bH , 118Bx, By, 88

C◦(P), 7C(P), 7C[u, v], 2

D, 26d∗, 83∆k(P, G), 8

E , 64E∗, 64ε, 10ε+, 10η, 65η, 12

Forb(S), 4Forb∗(S), 4

G, 2G, 6g, 4g, 4g, 4G∗, 40G+, 7g+, 65g+, 9g+, 80ga, 13g+a , 13G/xy, 6G◦xy, 6

Gxy, 6

H, 46, 110h0, 65h0, 11h0, 81h1, 65h1, 12h1, 81Hl, 23H(P), 70Hw, 70

L, 109l, 110λ(v), 45Lr, 113Lx, Ly, 88LXY, 46

mb, 119M(G), 4m(P,R), 119

Nk, 3nH , 117

pH , 117φ, 101Π, 3

RAB, 118ρH , 117

S, 70Sk, 3σ, 80σ+, 80sum(b), 118

t, 117τ(Q), τ(H), 110

INDEX OF SYMBOLS 130

τΠ, 3θ, 65θ, 81θ, 9Ti, 39T ′i , 39

v0, 110

X,Y, 45

INDEX OF DEFINITIONS 131

Index of Definitions

admitting linkages, 93alternating

k-alternating, 46xy-alternating, 10Π-angle, 3attach

bridge, 2attached

label, 47tripod, 5

attachments, 2avoid, 2

bar, 25base, 88block, 9X-block, 46boundary

of a pre-K-graph, 87of XY-labelled graph, 46

boundary cycle, 46boundary cycle of a core, 91branch, 2φ-branch, 101branch vertex, 2H-bridge, 2φ-bridge, 101v-bridge, 114C-bridge set, 52bridge vector, 118

cactus, 112cascade, 702-cell, 3center of labelled graph, 110U -completion, 91k-connected, 63-connected modulo K, 42connectivity, 6contain

label, 47

label sequence, 47Π-contractible cycle, 84core, 91covered, 46P-critical, 7critical class, 7φ-cross, 103cutedge, 6cutting

along a curve, 40of a vertex, 40

cutvertex, 6

dart, 3decrease, 8descriptor of a rooted cactus, 117disjoint xy-K-graphs, 87dumbbell, 25

edgecontraction, 4deletion, 4

Π-embedded, 3embedding, 3

combinatorial, 3nonorientable, 4orientable, 4

equivalent labelled graphs, 112equivalent rooted cacti, 114essentially 3-connected, 104Eulerian multiset of edges, 122

Π-face, 3φ-face, 103face-distance, 83Π-facial cycle, 3Π-facial walk, 3flipped embedding, 114foot

of a linkage, 91of a tripod, 5

INDEX OF DEFINITIONS 132

free φ-cross, 103fundamental cycle, 84

genus, 4alternating, 13Euler, 4nonorientable, 4orientable, 4

graph parameter, 7

homeomorphic embedding, 101hopper of level k, 23P -hopper, 70

induced subgraph, 122interior

of a bridge, 2of a core, 91of a pre-K-graph, 87

isomorphic, 6

φ-jump, 103

K-graph, 5x-K-graph, 87Kuratowski graph, 1Kuratowski subgraph, 4

label sequence, 46, 110label transition, 46, 110labelled graph, 109labelled vertex, 45XY-labelled graph, 45leaf-labelled cactus, 112U -linkage, 91U -linked, 91local bridge, 2

Min-Trans, 110AB-minimal sequence, 116minimum number of label transitions, 117minor, 4minor-monotone, 7minor-operation, 4

set, 4

nonseparating, 6number of transitions, 46, 110

obstruction, 4topological, 4

Π-one-sided, 3one-sided curve, 40ordering of types in b, 119orientably simple, 4orientizing

curve, 40face-width, 40

origin of a label, 110overlap, 2overlap graph, 2

parameter, see graph parameterparts, 6path

crossing, 5C-path, 5φ-path, 1033-path-condition, 84planar minor, 99pre-K-graph, 86present color, 124

removable foot, 91root of a cactus, 113rooted label sequence, 114rooted leaf-labelled cactus, 114

segment, 2r-separated, 8k-separated xy-K-graphs, 91separated vertices, 83(u, v)-separating cycle, 83separation of xy-K-graphs, 92B-side, 83γ-side, 40v-side, 83signature, 3singular, 3skew-overlap, 2

INDEX OF DEFINITIONS 133

splitting, 40splitting of xy-K-graphs, 91subdivision, 2xy-sum, 6sum of a bridge vector, 1182-sum, 6support, 2surface, 3

nonorientable, 3orientable, 3

terminal, 6threshold, 7P-tight, 7traversal permutation, 3φ-triad, 103tripod, 5φ-tripod, 104trivial bridge, 2Π-two-sided, 3type of a rooted cactus, 117

weak hopper, 70weight of a walk, 121width, 40