052180230X_Graph

Topics in Topological Graph Theory

The use of topological ideas to explore various aspects of graph theory, and vice versa, is a fruitfularea of research. There are links with other areas of mathematics, such as design theory and geometry,and increasingly with such areas as computer networks where symmetry is an important feature.Other books cover portions of the material here, but there are no other books with such a wide scope.

This book contains fifteen expository chapters written by acknowledged international experts inthe field. Their well-written contributions have been carefully edited to enhance readability and tostandardize the chapter structure, terminology and notation throughout the book. To help the reader,there is an extensive introductory chapter that covers the basic background material in graph theoryand the topology of surfaces. Each chapter concludes with an extensive list of references.

lowell w. beineke is Schrey Professor of Mathematics at Indiana UniversityPurdue UniversityFort Wayne, where he has been since receiving his Ph.D. from the University of Michigan under theguidance of Frank Harary. His graph theory interests are broad, and include topological graph theory,line graphs, tournaments, decompositions and vulnerability. With Robin Wilson he edited SelectedTopics in Graph Theory (3 volumes), Applications of Graph Theory, Graph Connections and Topicsin Algebraic Graph Theory. Until recently he was editor of the College Mathematics Journal.

robin j. wilson is Professor of Pure Mathematics at The Open University, UK, and EmeritusProfessor of Geometry at Gresham College, London. After graduating from Oxford, he received hisPh.D. in number theory from the University of Pennsylvania. He has written and edited many bookson graph theory and the history of mathematics, including Introduction to Graph Theory and FourColours Suffice, and his research interests include graph colourings and the history of combinatorics.He has won a Lester Ford Award and a George Polya Award from the MAA for his expository writing.

jonathan l. gross, Professor of Computer Science at Columbia University, served as anacademic consultant for this volume. His mathematical work in topology and graph theory haveearned him an Alfred P. Sloan Fellowship, an IBM Postdoctoral Fellowship, and numerous researchgrants. With Thomas Tucker, he wrote Topological Graph Theory and several fundamentalpioneering papers on voltage graphs and on enumerative methods. He has written and edited eightbooks on graph theory and combinatorics, seven books on computer programming topics, and onebook on cultural sociometry.

thomas w. tucker, Charles Hetherington Professor of Mathematics at Colgate University, alsoserved as an academic consultant for this volume. He has been at Colgate University since 1973, aftera Ph.D. in 3-manifolds from Dartmouth in 1971 and a post-doctoral position at Princeton. He isco-author (with Jonathan Gross) of Topological Graph Theory. His early publications were onnon-compact 3-manifolds, then topological graph theory, but his recent work is mostly algebraic,especially distinguishability and the group-theoretic structure of symmetric maps.

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

All the titles listed below can be obtained from good booksellers or from CambridgeUniversity Press. For a complete series listing visithttp://www.cambridge.org/uk/series/sSeries.asp?code=EOM

68 R. Goodman and N. R. Wallach Representations and Invariants of the Classical Groups69 T. Beth, D. Jungnickel, and H. Lenz Design Theory I, 2nd edn70 A. Pietsch and J. Wenzel Orthonormal Systems for Banach Space Geometry71 G. E. Andrews, R. Askey and R. Roy Special Functions72 R. Ticciati Quantum Field Theory for Mathematicians73 M. Stern Semimodular Lattices74 I. Lasiecka and R. Triggiani Control Theory for Partial Differential Equations I75 I. Lasiecka and R. Triggiani Control Theory for Partial Differential Equations II76 A. A. Ivanov Geometry of Sporadic Groups I77 A. Schinzel Polynomials with Special Regard to Reducibility78 T. Beth, D. Jungnickel and H. Lenz Design Theory II, 2nd edn79 T. W. Palmer Banach Algebras and the General Theory of *-Albegras II80 O. Stormark Lies Structural Approach to PDE Systems81 C. F. Dunkl and Y. Xu Orthogonal Polynomials of Several Variables82 J. P. Mayberry The Foundations of Mathematics in the Theory of Sets83 C. Foias, O. Manley, R. Rosa and R. Temam NavierStokes Equations and Turbulence84 B. Polster and G. Steinke Geometries on Surfaces85 R. B. Paris and D. Kaminski Asymptotics and MellinBarnes Integrals86 R. McEliece The Theory of Information and Coding, 2nd edn87 B. A. Magurn An Algebraic Introduction to K-Theory88 T. Mora Solving Polynomial Equation Systems I89 K. Bichteler Stochastic Integration with Jumps90 M. Lothaire Algebraic Combinatorics on Words91 A. A. Ivanov and S. V. Shpectorov Geometry of Sporadic Groups II92 P. McMullen and E. Schulte Abstract Regular Polytopes93 G. Gierz et al. Continuous Lattices and Domains94 S. R. Finch Mathematical Constants95 Y. Jabri The Mountain Pass Theorem96 G. Gasper and M. Rahman Basic Hypergeometric Series, 2nd edn97 M. C. Pedicchio and W. Tholen (eds.) Categorical Foundations98 M. E. H. Ismail Classical and Quantum Orthogonal Polynomials in One Variable99 T. Mora Solving Polynomial Equation Systems II

100 E. Olivieri and M. Eulalia Vares Large Deviations and Metastability101 A. Kushner, V. Lychagin and V. Rubtsov Contact Geometry and Nonlinear Differential Equations102 L. W. Beineke and R. J. Wilson (eds.) with P. J. Cameron Topics in Algebraic Graph Theory103 O. Staffans Well-Posed Linear Systems104 J. M. Lewis, S. Lakshmivarahan and S. K. Dhall Dynamic Data Assimilation105 M. Lothaire Applied Combinatorics on Words106 A. Markoe Analytic Tomography107 P. A. Martin Multiple Scattering108 R. A. Brualdi Combinatorial Matrix Classes110 M.-J. Lai and L. L. Schumaker Spline Functions on Triangulations111 R. T. Curtis Symmetric Generation of Groups112 H. Salzmann, T. Grundhofer, H. Hahl and R. Lowen The Classical Fields113 S. Peszat and J. Zabczyk Stochastic Partial Differential Equations with Levy Noise114 J. Beck Combinatorial Games116 D. Z. Arov and H. Dym J-Contractive Matrix Valued Functions and Related Topics117 R. Glowinski, J.-L. Lions and J. He Exact and Approximate Controllability for Distributed Parameter

Systems118 A. A. Borovkov and K. A. Borovkov Asymptotic Analysis of Random Walks119 M. Deza and M. Dutour Sikiric Geometry of Chemical Graphs120 T. Nishiura Absolute Measurable Spaces121 M. Prest Purity, Spectra and Localisation122 S. Khrushchev Orthogonal Polynomials and Continued Fractions: From Eulers Point of View123 H. Nagamochi and T. Ibaraki Algorithmic Aspects of Graph Connectivity124 F. W. King Hilbert Transforms I125 F. W. King Hilbert Transforms II126 O. Calin and D.-C. Chang Sub-Riemannian Geometry127 M. Grabisch, J.-L. Marichal, R. Mesiar and E. Pap Aggregation Functions

Leonhard Euler (17071783),the founder of topological graph theory.

Topics in Topological Graph Theory

Edited by

LOWELL W. BEINEKEIndiana UniversityPurdue University

Fort Wayne

ROBIN J. WILSONThe Open University

Academic Consultants

JONATHAN L. GROSSColumbia University

THOMAS W. TUCKERColgate University

cambridge university pressCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.orgInformation on this title: www.cambridge.org/9780521802307

c Cambridge University Press 2009

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place withoutthe written permission of Cambridge University Press.

First published 2009

Printed in the United Kingdom at the University Press, Cambridge

A catalogue record for this publication is available from the British Library

ISBN 978-0-521-80230-7 hardback

Cambridge University Press has no responsibility for the persistence oraccuracy of URLs for external or third-party internet websites referred to

in this publication, and does not guarantee that any content on suchwebsites is, or will remain, accurate or appropriate.

This book is dedicated to the memory of Gerhard Ringel (19192008),one of the pioneers of modern topological graph theory.

Contents

Foreword by Jonathan L. Gross and Thomas W. Tucker page xvPreface xvii

Introduction 1LOWELL W. BEINEKE and ROBIN J. WILSON1. Graph theory 12. Graphs in the plane 103. Surfaces 124. Graphs on surfaces 14

1 Embedding graphs on surfaces 18JONATHAN L. GROSS and THOMAS W. TUCKER1. Introduction 182. Graphs and surfaces 193. Embeddings 204. Rotation systems 235. Covering spaces and voltage graphs 266. Enumeration 297. Algorithms 308. Graph minors 31

2 Maximum genus 34JIANER CHEN and YUANQIU HUANG1. Introduction 342. Characterizations and complexity 363. Kuratowski-type theorems 384. Upper-embeddability 395. Lower bounds 40

ix

x Contents

3 Distribution of embeddings 45JONATHAN L. GROSS1. Introduction 452. Enumerating embeddings by surface type 483. Total embedding distributions 514. Congruence classes 535. The unimodality problem 556. Average genus 567. Stratification of embeddings 59

4 Algorithms and obstructions for embeddings 62BOJAN MOHAR1. Introduction 622. Planarity 643. Outerplanarity and face covers 664. Disc embeddings and the 2-path problem 685. Graph minors and obstructions 696. Algorithms for embeddability in general surfaces 737. Computing the genus 75

5 Graph minors: generalizing Kuratowskis theorem 81R. BRUCE RICHTER1. Introduction 812. Graph decompositions 843. Linked decompositions 884. Graphs with bounded tree-width 945. Finding large grids 996. Embedding large grids 107

6 Colouring graphs on surfaces 111JOAN P. HUTCHINSON1. Introduction 1112. High-end colouring 1133. A transition from high-end to low-end colouring 1164. Colouring graphs with few colours 1195. Girth and chromatic number 1246. List-colouring graphs 1257. More colouring extensions 1278. An open problem 129

Contents xi

7 Crossing numbers 133R. BRUCE RICHTER and G. SALAZAR1. Introduction 1332. What is the crossing number? 1353. General bounds 1374. Applications to geometry 1395. Crossing-critical graphs 1396. Other families of graphs 1437. Algorithmic questions 1448. Drawings in other surfaces 1469. Conclusion 147

8 Representing graphs and maps 151TOMAZ PISANSKI and ARJANA ZITNIK1. Introduction 1512. Representations of graphs 1523. Energy and optimal representations 1554. Representations of maps 1635. Representations of maps in the plane 1706. Representations of incidence geometries and related topics 174

9 Enumerating coverings 181JIN HO KWAK and JAEUN LEE1. Introduction 1812. Graph coverings 1833. Regular coverings 1854. Surface branched coverings 1905. Regular surface branched coverings 1936. Distribution of surface branched coverings 1957. Further remarks 196

10 Symmetric maps 199JOZEF SIRAN and THOMAS W. TUCKER1. Introduction 1992. Representing maps algebraically 2003. Regular maps 2054. Cayley maps 2105. Regular Cayley maps 2126. Edge-transitive maps 2187. Maps and mathematics 221

xii Contents

11 The genus of a group 225THOMAS W. TUCKER1. Introduction 2252. Symmetric embeddings and groups acting on surfaces 2263. Quotient embeddings and voltage graphs 2284. Inequalities 2325. Groups of low genus 2356. Genera of families of groups 239

12 Embeddings and geometries 245ARTHUR T. WHITE1. Introduction 2452. Surface models 2483. Projective geometries 2504. Affine geometries 2535. 3-configurations 2566. Partial geometries 2607. Regular embeddings for PG(2, n) 2648. Problems 265

13 Embeddings and designs 268M. J. GRANNELL and T. S. GRIGGS1. Introduction 2682. Steiner triple systems and triangulations 2703. Recursive constructions 2734. Small systems 2785. Cyclic embeddings 2806. Concluding remarks 284

14 Infinite graphs and planar maps 289MARK E. WATKINS1. Introduction 2892. Ends 2903. Automorphisms 2934. Connectivities 2955. Growth 3006. Infinite planar graphs and maps 303

Contents xiii

15 Open problems 313DAN ARCHDEACON

1. Introduction 3132. Drawings and crossings 3143. Genus and obstructions 3174. Cycles and factors 3205. Colourings and flows 3226. Local planarity 3247. Thickness, book embeddings and covering graphs 3258. Geometrical topics 3289. Algorithms 330

10. Infinite graphs 332

Notes on contributors 337Index 341

Foreword

The origins of topological graph theory lie in the 19th century, largely with thefour colour problem and its extension to higher-order surfaces the Heawoodmap problem. With the explosive growth of topology in the early 20th century,mathematicians like Veblen, Rado and Papakyriakopoulos provided foundationalresults for understanding surfaces combinatorially and algebraically. Kuratowski,MacLane and Whitney in the 1930s approached the four colour problem as aquestion about the structure of graphs that can be drawn without edge-crossingsin the plane. Kuratowskis theorem characterizing planarity by two obstructionsis the most famous, and its generalization to the higher-order surfaces became aninfluential unsolved problem.

The second half of the 20th century saw the solutions of all three problems: theHeawood map problem by Ringel, Youngs et al. by 1968, the four colour problemby Appel and Haken in 1976, and finally the generalized Kuratowski problem byRobertson and Seymour in the mid-1990s. Each is a landmark of 20th-centurymathematics. The RingelYoungs work led to an alliance between combinatoricsand the algebraic topology of branched coverings. The AppelHaken work was thefirst time that a mathematical theorem relied on exhaustive computer calculations.And the RobertsonSeymour work led to their solution of Wagners conjecture,which provides a breathtaking structure for the collection of all finite graphs, acollection that would seem to have no structure at all.

Each of these problems centres on the question of which graphs can beembedded in which surfaces, with two complementary perspectives fixing thegraph or fixing the surface. Although the question sounds highly focused, the studyof graphs on surfaces turns out to be incredibly broad, rich in connections with otherbranches of mathematics and computer science: algorithms, computer-drawing,group theory, Riemann surfaces, enumerative combinatorics, block designs, finitegeometries, Euclidean and non-Euclidean geometry, knot theory, the absoluteGalois group, C*-algebras, and even string theory.

xv

xvi Foreword

This volume attempts to survey the principal results within over-arching themesfor the myriad aspects of topological graph theory. The authors of the chapters arerecognized authorities in their fields. This book is written for the non-specialist andcan be used as the basis for a graduate-level course. Nonetheless, the individualchapters cover their fields in great depth and detail, so that even specialists will findthe book valuable, both as a reference and as a source of new insights and problems.

JONATHAN L. GROSSTHOMAS W. TUCKER

Preface

The field of graph theory has undergone tremendous growth during the past century.As recently as fifty years ago, the graph theory community had few membersand most were in Europe and North America; today there are hundreds of graphtheorists and they span the globe. By the mid-1970s, the field had reached thepoint where we perceived the need for a collection of surveys of the areas ofgraph theory: the result was our three-volume series Selected Topics in GraphTheory, comprising articles written by distinguished experts in a common style.During the past quarter-century, the transformation of the subject has continued,with individual areas (such as topological graph theory) expanding to the point ofhaving important sub-branches themselves. This inspired us to conceive of a newseries of books, each a collection of articles within a particular area written byexperts within that area. The first of these books was our companion volume onalgebraic graph theory, published in 2004. This is the second of these books.

One innovative feature of these volumes is the engagement of academicconsultants (here, Jonathan Gross and Thomas Tucker) to advise us on topicsto be included and authors to be invited. We believe that this has been successful,the result being chapters covering the full range of areas within topological graphtheory written by authors from around the world. Another important feature is thatwe have imposed uniform terminology and notation throughout, as far as possible,in the belief that this will aid readers in going from one chapter to another. For asimilar reason we have not attempted to remove a small amount of overlap betweenthe various chapters.

We hope that these features will make the book easier to use in an advancedcourse or seminar. We heartily thank the authors for cooperating on this,even though it sometimes required their abandoning some of their favouriteconventions for example, many mathematicians use to denote the Eulercharacteristic, whereas for graph theorists usually denotes the chromatic number:the graph theorists won on this one. We also asked our contributors to undergo

xvii

xviii Preface

the ordeal of having their early versions subjected to detailed critical reading. Webelieve that the final product is thereby significantly better than it might otherwisehave been, simply a collection of individually authored chapters. We want toexpress our sincere appreciation to all of our contributors for their cooperation.

We extend special thanks to Jonathan Gross and Thomas Tucker for theirwillingness to share their expertise as academic consultants their advice has beeninvaluable. We are also grateful to Cambridge University Press for publishing thiswork; in particular, we thank Roger Astley and Clare Dennison for their advice,support and cooperation. Finally, we extend our appreciation to several universitiesfor the different ways in which they have assisted with this endeavour: the firsteditor is grateful to Indiana UniversityPurdue University in Fort Wayne, while thesecond editor has had the cooperation of the Open University and Keble College,Oxford.

LOWELL W. BEINEKEROBIN J. WILSON

IntroductionLOWELL W. BEINEKE and ROBIN J. WILSON

1. Graph theory2. Graphs in the plane3. Surfaces4. Graphs on surfacesReferences

1. Graph theoryThis section presents the basic definitions, terminology and notation of graphtheory, along with some fundamental results. Further information can be foundin the many standard books on the subject for example, Chartrand and Lesniak[1], Gross and Yellen [2], West [3] or (for a simpler treatment) Wilson [4].

GraphsA graph G is a pair of sets (V ,E), where V is a finite non-empty set of elementscalled vertices, and E is a finite set of elements called edges, each of which has twoassociated vertices (which may be the same). The sets V and E are the vertex-setand edge-set of G, and are sometimes denoted by V (G) and E(G). The orderof G is the number of vertices, usually denoted by n, and the number of edges isdenoted by m.

An edge whose vertices coincide is called a loop, and if two vertices are joinedby more than one edge, these are called multiple edges. A graph with no loopsor multiple edges is a simple graph. In many areas of graph theory there is littleneed for graphs that are not simple, in which case an edge e can be considered asa pair of vertices, e = {v, w}, or vw for simplicity. However, in topological graphtheory, it is often useful, and sometimes necessary, to allow loops and multiple

1

2 Lowell W. Beineke and Robin J. Wilson

Fig. 1.

edges. A graph of order 4 and its underlying simple graph are shown in Fig. 1. Thecomplement G of a simple graph G has the same vertices as G, but two verticesare adjacent in G if and only if they are not adjacent in G.

Adjacency and degreesThe vertices of an edge are incident with the edge, and the edge is said to join thesevertices. Two vertices that are joined by an edge are neighbours and are said tobe adjacent. The set N(v) of neighbours of a vertex v is its neighbourhood. Twoedges are adjacent if they have a vertex in common.

The degree deg v of a vertex v is the number of times that it occurs as anendpoint of an edge (with a loop counted twice); in a simple graph, the degree ofa vertex is just the number of its neighbours. A vertex of degree 0 is an isolatedvertex and one of degree 1 is an end vertex. A graph is regular if all of its verticeshave the same degree, and is k-regular if that degree is k; a 3-regular graph issometimes called cubic. The maximum degree in a graph G may be denoted by or (G), and the minimum degree by or (G).

Isomorphisms and automorphismsAn isomorphism between two graphs G and H consists of a pair of bijections,one between their vertex-sets and the other between their edge-sets, that preserveincidence and non-incidence. (For simple graphs, this amounts to having a bijectionbetween their vertex-sets that preserves adjacency and non-adjacency.) The graphsG and H are isomorphic, denoted by G H or G = H , if there exists anisomorphism between them.

An automorphism of a graph G is an isomorphism of G with itself. The set ofall automorphisms of a graph G forms a group, called the automorphism groupof G and denoted by Aut(G). A graph G is vertex-transitive if, for any verticesv and w, there is an automorphism taking v to w. It is edge-transitive if, for anyedges e and f , there is an automorphism taking the vertices of e to those of f .It is arc-transitive if, given two ordered pairs of adjacent vertices (v, w) and

Introduction 3

(v, w), there is an automorphism taking v to v and w to w. This is stronger thanedge-transitivity, since it implies that for each edge there is an automorphism thatinterchanges its vertices.

Walks, paths and cyclesA walk in a graph is a sequence of vertices and edges v0, e1, v1, . . . , ek , vk , inwhich each edge ei joins the vertices vi1 and vi . This walk goes from v0 to vk orconnects v0 and vk , and is called a v0-vk walk. For simple graphs, it is frequentlyshortened to v0v1 vk , since the edges can be inferred from this. Its length is k,the number of occurrences of edges, and if v0 = vk , the walk is closed. Someimportant types of walk are the following:

a path is a walk in which no vertex is repeated; a cycle is a non-trivial closed walk in which no vertex is repeated, except the

first and last; a trail is a walk in which no edge is repeated; a circuit is a non-trivial closed trail.

Connectedness and distanceA graph is connected if there is a path connecting each pair of vertices, anddisconnected otherwise. A (connected ) component of a graph is a maximalconnected subgraph.

In a connected graph, the distance d(v, w) from v to w is the length of a shortestv-w path. It is easy to see that distance satisfies the properties of a metric.

The diameter of a connected graph G is the maximum distance between twovertices of G. If G has a cycle, the girth of G is the length of a shortest cycle.

A connected graph is Eulerian if it has a closed trail containing all the edges ofG; such a trail is an Eulerian trail. The following are equivalent for a connectedgraph G:

G is Eulerian; the degree of each vertex of G is even; the edge-set of G can be partitioned into cycles.

A graph is Hamiltonian if it has a spanning cycle, and is traceable if it has aspanning path. No good characterizations of these properties are known.

Bipartite graphs and treesIf the set of vertices of a graph G can be partitioned into two non-empty subsetsso that no edge joins two vertices in the same subset, then G is bipartite. The two


subsets are called partite sets and, if they have orders r and s, G is sometimescalled an r s bipartite graph. (For convenience, the graph with one vertex andno edges is also called bipartite.) Bipartite graphs are characterized by having nocycles of odd length.

Among the bipartite graphs are trees, those connected graphs with no cycles.Trees have been characterized in many ways, some of which we give here. For agraph of order n, the following statements are equivalent:

G is connected and has no cycles; G is connected and has n 1 edges; G has no cycles and has n 1 edges; G has exactly one path between any two vertices.

A graph without cycles is called a forest; thus, each component of a forest is a tree.The set of trees can also be defined inductively: a single vertex is a tree; and

for n 1, the trees with n+1 vertices are those graphs obtainable from some treewith n vertices by adding a new vertex adjacent to precisely one of its vertices.

This definition has a natural extension to higher dimensions. The k-dimensionaltrees, or k-trees for short, are defined as follows. The complete graph on k verticesis a k-tree, and for n k, the k-trees with n+1 vertices are those graphs obtainablefrom some k-tree with n vertices by adding a new vertex adjacent to k mutuallyadjacent vertices in the k-tree. Fig. 2 shows a tree and a 2-tree. An importantconcept in the study of graph minors (introduced later) is the tree-width of a graphG, the minimum dimension of any k-tree that contains G as a subgraph.

Fig. 2.

Special graphsWe now introduce some individual types of simple graph:

the complete graph Kn has n vertices, each adjacent to all the others; the path graph Pn consists of the vertices and edges of a path of length n 1; the cycle graph Cn consists of the vertices and edges of a cycle of length n;

Introduction 5

the complete bipartite graph Kr,s is the simple r s bipartite graph in whicheach vertex is adjacent to every vertex in the other partite set;

in the complete k-partite graph Kr1,r2,...,rk the vertices are in k sets with ordersr1, r2, . . ., rk , and each vertex is adjacent to every vertex in another set; if the ksets all have order r , the graph is denoted by Kk(r).

Examples of these graphs are given in Fig. 3.

K5:P5: C5:

K3(2):K3,3:

Fig. 3.

We also introduce some special graphs that are not simple:

the bouquet Bm has one vertex and m incident loops; the dipole Dm consists of two vertices with m edges joining them; the cobblestone path is the 4-regular graph obtained from the pathPn by doubling

each edge and adding a loop at each end.

Fig. 4 gives examples of these graphs.

B4: D4: J3:

Fig. 4.

A necklace is any graph obtained from a cycle by doubling each edge in anindependent subset of its edges and adding a loop at each vertex that is not on


Fig. 5.

one of those edges. It is of type (r, s) if it has r doubled edges and s loops (so theoriginal cycle has length 2r + s). The necklace in Fig. 5 is of type (3, 4).

OperationsLet G and H be graphs with disjoint vertex-sets V (G) = {v1, v2, . . . , vn} andV (H) = {w1, w2, . . . , wr}: the union GH has vertex-set V (G)V (H) and edge-set E(G)E(H). The

union of k graphs isomorphic to G is denoted by kG; the join G + H is obtained from G H by adding an edge from each vertex in

G to each vertex in H ; the Cartesian product GH (or G H ) has vertex-set V (G) V (H), and

(vi , wj) is adjacent to (vh, wk) if either vi is adjacent to vh in G and wj = wk ,or vi = vh and wj is adjacent to wk in H : in less formal terms, GH can beobtained by taking n copies of H and joining corresponding vertices in differentcopies whenever there is an edge in G;

the lexicographic product (or composition) G[H ] also has vertex-set V (G) V (H), but with (vi, wj ) adjacent to (vh,wk) if either vi is adjacent to vh in Gor vi = vh and wj is adjacent to wk in H .

Examples of these binary operations are given in Fig. 6.

Subgraphs and minorsIf G and H are graphs with V (H) V (G) and E(H) E(G), then H is asubgraph of G; if, moreover, V (H) = V (G), then H is a spanning subgraph. The

Introduction 7

G:

H:

G H: G + H:

G H: G[H]:

Fig. 6.

subgraph S induced by a non-empty set of S of vertices of G is the subgraph Hwhose vertex-set is S and whose edge-set consists of those edges of G that jointwo vertices in S. A subgraph H of G is an induced subgraph if H = V (H). InFig. 7, H1 is a spanning subgraph of G, and H2 is an induced subgraph.

Graph

G: H1: H2:

Spanning subgraph Induced subgraph

Fig. 7.

Given a graph G, the deletion of a vertex v results in the subgraph obtained byremoving v and all of its incident edges; it is denoted by Gv and is the subgraphinduced by V {v}. More generally, if S is any set of vertices in G, then G Sis the graph obtained from G by deleting all the vertices in S and their incident


edges that is, G S = V S. Similarly, the deletion of an edge e results inthe subgraph G e and, for any set X of edges, GX is the graph obtained fromG by deleting all the edges in X.

There are two other operations that are especially important for topologicalgraph theory. If an edge e joins vertices v and w, the subdivision of e replaces e bya new vertex u and two new edges vu and uw. Two graphs are homeomorphic ifthere is some graph from which each can be obtained by a sequence of subdivisions.The contraction of e replaces the vertices v and w of e by a new vertex u, with anedgeux for each edge vx orwx inG. The operations of subdivision and contractionare illustrated in Fig. 8.

v

e

w

u

ContractionSubdivision

u

v

w

Fig. 8.

If a graph H can be obtained from G by sequence of edge-contractions and theremoval of isolated vertices, then G is contractible to H .Aminor of G is any graphthat can be obtained fromGby a sequence of edge-deletions and edge-contractions,along with deletions of isolated vertices.

Connectedness and connectivityA vertex v of G is a cut-vertex if G v has more components than G. A non-trivial connected graph with no cut-vertices is 2-connected or non-separable. Thefollowing statements are equivalent for a graph G with at least three vertices:

G is non-separable; every pair of vertices lie on a cycle; for any three vertices u, v and w, there is a u-w path containing v; for any three vertices u, v and w, there is a u-w path not containing v.

Introduction 9

More generally, G is k-connected if there is no set S of fewer than k vertices forwhichGS is a connected non-trivial graph. Menger gave a useful characterizationof such graphs:

Mengers theorem (vertex version) A graph G is k-connected if and only if, foreach pair of vertices v and w, there is a set of k internally disjoint v-w paths.

The connectivity (G) of a graph G is the maximum value of k for which G isk-connected.

There are similar concepts and results for edges. A cut-edge (or bridge) is anedge whose deletion produces one more component than before. (Note: for someauthors, bridgehas a different meaning.) A non-trivial graph is k-edge-connectedif the result of removing fewer than k edges is always connected, and the edge-connectivity (G) is the maximum value of k for which G is k-edge-connected.We note that Mengers theorem also has an edge version:

Mengers Theorem (edge version) A graph G is k-edge-connected if and only if,for each pair of vertices v and w, there is a set of k edge-disjoint v-w paths.

Graph colouringsAgraph is k-colourable if, from a set of k colours, it is possible to assign a colour toeach vertex in such a way that adjacent vertices always have different colours. Thechromatic number (G) is the least value of k for which G is k-colourable, andif (G) = k, then G is k-chromatic. It is easy to see that a graph is 2-colourableif and only if it is bipartite, but there is no good way to determine which graphsare k-colourable, for any k 3. Brookss theorem provides one of the best-knownbounds on the chromatic number of a graph.

Brookss theorem If G is a simple graph with maximum degree and is neitheran odd cycle nor a complete graph, then (G) .

There are similar concepts for colouring edges. A graph is k-edge-colourableif, from a set of k colours, it is possible to assign a colour to each edge in such away that adjacent edges always have different colours. The chromatic index (G)is the least value of k for which G is k-edge-colourable. Vizing [11] proved thatthe range of values of (G) is quite limited:

Vizings theorem If G is a simple graph with maximum degree , then (G) = or + 1.


Directed graphsDigraphs are directed analogues of graphs, and thus have many similarities, aswell as some important differences.

A digraph (or directed graph) D is a pair of sets (V ,E), where V is a finite non-empty set of elements called vertices, and E is a set of ordered pairs of distinctelements of V called arcs or directed edges. Note that the elements of E areordered, which gives each of them a direction. An example of a digraph is shownin Fig. 9.

D:

v2

v1 v4

v3

Fig. 9.

Because of the similarities between graphs and digraphs, we mention only themain differences here and do not redefine those concepts that carry over easily. Anarc (v,w) in a simple digraph may be written as vw, and is said to go from v tow, or to go out of v and into w. In the context of digraphs, walks, paths, cycles,trails and circuits are understood to be directed, unless otherwise indicated.

A digraph D is strongly connected if there is a path from each vertex to eachof the others. A strong component is a maximal strongly connected subgraph.Connectivity and edge-connectivity are defined in terms of strong connectedness.

2. Graphs in the planeIn this section we briefly survey properties of graphs that can be drawn in the planewithout any edges crossing. To make this more precise, we define an embeddingof a graph G in the plane to be a one-to-one mapping of the vertices of G into theplane and a mapping of the edges of G to disjoint simple open arcs, so that theimage of each edge joins the images of its two vertices and none of the images ofthe edges contains the image of a vertex.

Here there is little to be gained by allowing loops or multiple edges, so in thissection we assume that all graphs are simple. A graph that can be embedded in theplane is called a planar graph, and its image is called a plane graph. An exampleis given in Fig. 10.

A region of an embedded graph G is a maximal connected set of points in therelative complement of G in the plane; note that one region is unbounded. Thetopological closure of a region (that is, the region together with the vertices and

Introduction 11

r4r1

r2

r3

Fig. 10.

edges of G on its boundary) is a face. If a face has a connected boundary, thatboundary is a closed walk, and the length of that walk is the size of the face.

Steinitz showed that 3-connected planar graphs form a particularly nice classof graphs:

Steinitzs theorem A graph is the skeleton of a polyhedron if and only if it is3-connected and planar.

The fundamental theorem on planar graphs is an extension of Eulers well-known formula for polyhedra:

Eulers formula If an embedding of a connected graph in the plane has n vertices,m edges and r regions, then n m + r = 2.

One consequence of Eulers formula is that a planar graph G with n ( 3)vertices has at most 3(n2) edges, and at most 2(n2) if G is bipartite. It is easyto deduce from these observations that the graphs K5 and K3,3 are non-planar.Kuratowski proved that these are in fact the only barriers to planarity:

Kuratowskis theorem A graph is planar if and only if it does not contain asubgraph homeomorphic to K5 or K3,3.

There are other criteria for planarity, but we mention only one here; it is due toWagner and simply has the word contractible in place of homeomorphic:

Wagners theorem A graph is planar if and only if it does not contain a subgraphcontractible to K5 or K3,3.

There are several measures of non-planarity, two of which are as follows:

the crossing number of a graph G is (informally, but intuitively clear) theminimum number of crossings of pairs of edges in any drawing of G in theplane;

the thickness of G is the minimum number of graphs in a set of planar graphswhose union is G.


3. SurfacesMuch of the interest in topological graph theory involves graphs on surfaces otherthan the plane. In this section we say what is meant by a surface; the topic ofembeddability on these surfaces is introduced in the next section.

A surface is a topological space with the following two properties: it is a 2-manifold that is, each point has a neighbourhood homeomorphic to

an open disc; it is compact that is, it is closed and bounded.

Note that this definition is quite restrictive the plane does not qualify as a surfacein this sense since it is not compact, and the Mbius strip does not qualify since ithas a boundary and thus is not compact.

A surface is orientable if a positive sense of rotation (say, clockwise) can bemade around all points consistently, and is non-orientable otherwise. The simplestorientable surfaces are the sphere and the torus (Fig. 11), while the simplest non-orientable surfaces are the projective plane and the Klein bottle (Fig. 12).

Fig. 11.

Fig. 12.

There are two ways to increase the complexity of a surface: by adding eithera handle or a crosscap. To add a handle to a surface S, remove two disjoint opendiscs from S and identify their boundaries with the ends of a truncated cylinder in aconsistent manner (see Fig. 13). To add a crosscap to S, remove one open disc andidentify its boundary with that of a Mbius strip (see Fig. 14); this is equivalent toidentifying opposite points on the boundary of the disc.

It is a fact that no matter how h handles are added to the sphere, the resultis effectively the same: an orientable surface that we denote by Sh. Similarly, nomatter how k crosscaps (k > 0) are added to the sphere, the result is effectively the

Introduction 13

Fig. 13.

Fig. 14.

same: a non-orientable surface that we denote by Nk . Furthermore, every surfaceis homeomorphic either to Sh for some h 0, or to Nk for some k 1.

When the number of handles or crosscaps on a surface is small, it can be usefulto represent it as a polygon. For an orientable surface Sh, take a 4h-gon andidentify its sides according to the pattern

1, 1, 11 ,

11 , . . . , h, h,

1h ,

1h .

Fig. 15 shows the torus and the double torus represented in this way.

a

a

b bb2

b1

b1

a1b2

a2

a1

a2

Torus Double-torus

Fig. 15.


For the non-orientable surface Nk , take a 2k-gon and identify its sides accordingto the pattern

1, 1, 2, 2, . . . , k , k .

Fig. 16 shows the projective plane and the Klein bottle represented in this way.

Projective plane

a

a2 a1

a1a2

a

Klein bottle

Fig. 16.

4. Graphs on surfacesMany of the concepts we introduced in Section 2 carry over in a natural way tographs on surfaces in general. Specifically, a graph on a surface S is the analogueof a plane graph, and a graph G is embeddable on S if it is isomorphic to a graphon that surface; we refer to this as an embedding of G. An embedding is cellularif every region is homeomorphic to an open disc. Figs. 17 and 18 show cellularembeddings of the complete graphs K6 and K7 on the projective plane and thetorus.

Regions and faces are defined as for the plane. A cellular embedding in whicheach face has three sides is a triangulation, and one in which each face has foursides is a quadrangulation. (In this section all graphs are assumed to be simple.)The embeddings in Figs. 17 and 18 are both triangulations.

Every surface has a version of Eulers polyhedron formula:

Eulers formula If a simple graph G has a cellular embedding in a surface S withn vertices, m edges and r regions, then

n m + r ={

2 2h if S = Sh,2 k if S = Nk.

The number associated with S in this theorem is called its Euler characteristic,denoted by (S).

Introduction 15

a

b c

d

e

a

bc

d

e

Fig. 17.

a b c d a

a b c d a

e

f

g

e

f

g

Fig. 18.

The genus of a graphThe genus (G) of a graph G is the minimum genus of a surface in which thegraph can be embedded that is, the minimum number of handles that need tobe added to the sphere for G to be embeddable. It follows from Eulers formulathat (G) has the general lower bound of 16m 12n + 1; this can be improved to14m 12n + 1 if G is bipartite. In part because of the connection with colouringmaps on surfaces, much of the focus of early work was on the genus of completegraphs. The solution to this difficult problem was finally completed by Ringel andYoungs (see [7]) in 1968:

RingelYoungs theorem For n 3, the genus of the complete graph Kn is

(Kn) = 112 (n 3) (n 4).

An important consequence of this result is that it yields the chromatic number ofevery orientable surface other than the plane.


The crosscap number of a graphThe non-orientable analogue of the genus of a graph is the non-orientable genus(or crosscap number) (G), the minimum number of crosscaps that need to beadded to the sphere for G to be embeddable. In 1954, Ringel determined thenon-orientable genus of the complete graph:

Ringels theorem For n 3, the non-orientable genus of the complete graphKn is

(Kn) = 1

6 (n 3) (n 4),

except that (K7) = 3.There are some interesting comparisons to be made between the parameters

(G) and (G). Since any surface with a crosscap is non-orientable, it followsthat, for any graph G, (G) 2 (G) + 1. There is no bound in the otherdirection, however, as there are graphs of arbitrarily large orientable genus thatcan be embedded in the projective plane.

Other differences appear in cellular embeddings of graphs. If (G) = h, thenevery embedding of G on Sh is cellular, but the corresponding statement for thenon-orientable genus does not hold. In particular, although (K7) = 3, not everyembedding of K7 in N3 is cellular. Furthermore, while the orientable genus isadditive over the blocks of a graph the non-orientable genus is not the graphconsisting of two copies of K7 is a counter-example.

The chromatic number of a surfaceMuch of the interest in embedding complete graphs is related to colourings of mapsand graphs. The chromatic number (S) of a surface S is the maximum chromaticnumber among all S-embeddable graphs. As Heawood noted, a lower bound for(S) can be deduced from Eulers formula. For all surfaces other than the sphere,the sharpness of this bound follows from the genus and the non-orientable genusof complete graphs (and a little more for the Klein bottle).Map colour theorem Except for the Klein bottle N2, which has chromatic number6, the chromatic number of a surface S of Euler characteristic is

(S) = 12 (7 + 49 24) .Kuratowski-type theorems

Every surface has a family of graphs that plays the role of K5 and K3,3 inKuratowskis theorem. A minor-minimal forbidden family M(S) of a surface Sis a set of graphs with these three properties:

Introduction 17

no graph in M(S) is embeddable in S; every graph that is not embeddable in S has a graph in M(S) as a minor; no graph in M(S) is a minor of another graph in M(S).

There is a corresponding (and larger) family if, instead of minors, homeo-morphic subgraphs are considered. The projective plane is the only surface otherthan the sphere for which these families are known: the minor-minimal familycontains 35 graphs and the homeomorphically irreducible family contains 103.

One of the foremost results in topological graph theory is that these families arealways finite. For non-orientable surfaces this was established by Archdeacon andHuneke in 1980. The orientable case was not settled until 1984, when Robertsonand Seymour proved their spectacular result on graph minors.

RobertsonSeymour theorem Every infinite collection of graphs contains at leastone graph that is a minor of another.

Corollary The set of minor-minimal forbidden graphs of every surface is finite.

References1. G. Chartrand and L. Lesniak, Graphs and Digraphs (4th edn), Chapman & Hall/CRC,

2004.2. J. L. Gross and J. Yellen, Graph Theory and its Applications (2nd edn), Chapman &

Hall/CRC, 2005.3. D. B. West, Introduction to Graph Theory (3rd edn), Prentice-Hall, 2007.4. R. J. Wilson, Introduction to Graph Theory (4th edn), Pearson, 1996.

1Embedding graphs on surfaces

JONATHAN L. GROSS and THOMAS W. TUCKER

1. Introduction2. Graphs and surfaces3. Embeddings4. Rotation systems5. Covering spaces and voltage graphs6. Enumeration7. Algorithms8. Graph minorsReferences

In this first chapter, we review the basic ideas of topological graph theory.We describe the principal early theme of constructing embeddings, and wethen survey the launching of the dominant programmatic themes of thepresent era, which are presented in greater detail individually in subsequentchapters.

1. Introduction

By the late 19th century, the work of Heawood [16] and Heffter [17] had expandedthe study of graph drawings beyond the confines of the plane to surfaces of higherorder. Over the next hundred years or so, the solution of several long-standingproblems attracted many researchers and the present-day programmatic themeswere set into place. Of course, some of the methods used in the solutions led tonew problems. Topological graph theory is now one of the largest branches ofgraph theory.

This chapter gives a brief overview of some of the principal concepts,terminology and notation of topological graph theory. As general resources, werecommend [13], Chapter 7 of [14], [22] and [44].

18

1 Embedding graphs on surfaces 19

2. Graphs and surfacesWe start by recalling some definitions from the Introduction.Agraph G is formallydefined to be a combinatorial incidence structure with a vertex-set V and an edge-set E, where each edge e is incident with at most two vertices; we may write VGand EG, respectively, when there is more than one graph under consideration. Agraph may have multiple adjacencies and loops and is usually taken to be finiteunless the immediate context implies otherwise. In some contexts, the letters nand m are reserved for the number |V | of vertices and the number |E| of edges.

The underlying topological space of a graph, also commonly called the graph,is the 1-dimensional cell-complex with a 0-cell for each vertex v and a 1-cell foreach edge e. In that sense, each edge has two edge-ends and two endpoints (whichmay coincide if the edge is a loop). In some contexts, edges are assigned a directionfrom one endpoint to the other, usually indicated in a drawing by an arrow on anedge. The degree, also called the valence, of a vertex v is the number of edge-endsincident to v, that is the number of incident edges with loops weighted twice. Agraph is called regular if all vertices have the same degree.

A surface S is a connected topological space such that every point has an openneighborhood homeomorphic to the interior of the unit disc (this definition doesnot allow a surface to have boundary). A surface is closed if is compact, and isorientable if it contains no subset homeomorphic to the Mbius band (the spaceobtained by identifying a pair of opposite sides of a rectangle as shown in Fig. 1).Just as the plane has two orientations, clockwise and anticlockwise, an orientablesurface has two possible orientations; if one is specified, we say that the surfaceis oriented.

Fig. 1. Constructing a Mbius band

Every surface has a triangulation [27] into homeomorphic copies of a triangle,which intersect only along entire edges or at vertices. This so-called piecewise-linear structure of a surface as a 2-complex is unique up to piecewise-linearhomeomorphism [24], which means that surfaces can be treated as combinatorialobjects, as well as topological ones. The Introduction to this volume describes boththe orientable surfaces Sh and the non-orientable surfaces Nk .

20 Jonathan L. Gross and Thomas W. Tucker

3. EmbeddingsAn embedding of a graph G on a surface S is a continuous function f : G Staking G homeomorphically to its image f (G). Intuitively, an embedding is adrawing of a graph on a surface in which no edges cross. The components of thecomplement of the image of an embedded graph are called regions.

An embedding is cellular if every region is homeomorphic to an open disc. Innearly every aspect of topological graph theory, the only embeddings consideredare cellular. Consequently, the word cellular is usually omitted by most authors,as it will be in this book. Since an embedding of a non-connected graph cannot becellular, it is often implicit from the context that an embedded graph is connected.

For a cellular embedding, the topological closure of a region is called a face. Theset of faces is denoted by F , with subscripts as needed if more than one embeddingis under consideration. The closed walk in the underlying graph G correspondingto the boundary of a region is called a boundary walk or face boundary walk ofthe corresponding face, and is unique up to the choice of the initial vertex and thechoice of orientation of the region. The length of this boundary walk is called thesize of the face.

An embedding is strongly cellular or circular if every face is homeomorphic to aclosed disc that is, if every boundary walk is a cycle. If G is not 2-connected, thenit has no strongly cellular embedding. The converse result, that every 2-connectedgraph has a strongly cellular embedding, is known as the circular embeddingconjecture and is open at this time. It implies the closely related cycle doublecover conjecture (of [33] and [34]) that every 2-edge-connected graph has acollection of cycles that includes every edge exactly twice. Chapter 15 providesfurther information about these conjectures as part of a large collection of openproblems.

The orientable genus range of a graph G is the set of integers h (which areeasily proved to be consecutive) such that the graph G is cellularly embeddablein the surface Sh. The minimum of this range is called the minimum genus of thegraph (or often simply, the genus). The maximum is called the maximum genus.The minimum and maximum genus of the graph G are denoted by min(G) (oroften, simply (G)) and max(G), respectively. A graph of genus 0 is said to beplanar.

The crosscap range of a graph G is the set of integers k (also easily proved tobe consecutive) such that the graph G is cellularly embeddable in the surface Nk .The minimum of this range is called the minimum crosscap number of the graph(or, often, simply the crosscap number). The maximum is called the maximumcrosscap number. The minimum and maximum crosscap numbers of the graph Gare denoted by min(G) and max(G), respectively.


The Poincar dual embedding for a cellular graph embedding G S (calledthe primal embedding in this context) is constructed as follows: in the interior of each primal region, a dual vertex is drawn; through each primal edge, a dual edge is drawn joining the dual vertex on one

side of the edge to the dual vertex on the other (thus, a loop whenever the sameprimal region lies on both sides of that primal edge);

if the surfaceS is oriented, then in the dual embedding, the orientation is reversed.Aflat-polygon representation of an embeddingK4 S1 and its dual embedding

are shown in Fig. 2. The primal vertices and the primal edges are solid, the dualvertices are hollow, and the dual edges are dashed. Observe that four edges jointhe two dual vertices and that one of the dual vertices has two loops incident withit. There are many other ways to draw a graph embedding or even a graph (in3-space, with edge-crossings in the plane, with straight lines as edges, with ovalsas edges, etc). Chapter 8 surveys the different ways that one can try to visualize agraph or embedding.

b b

a

a

Fig. 2. A toroidal embedding and its dual

The Euler polyhedron formula for a cellular embedding of a graph is

|V | |E| + |F | ={

2 2h for the orientable surface Sh,2 k for the non-orientable surface Nk.

The value of the expression on the right side of the equation is called the Eulercharacteristic of the surface, denoted by .

The special case |V ||E|+|F | = 2 for the sphere, first stated by Euler in 1750,is reasonably regarded as the first result of topological graph theory, even thoughvarious topological aspects, such as the Jordan curve theorem, the Schoenfliesstheorem, and the triangulability of surfaces, were not proved until early in the 20thcentury. (There are many proofs; see, for example, [13].)


Amap on a surface S is another name for an embeddingG S of a graph into Sand is used when the focus is on the symmetries of the underlying vertexedgefaceincidence structure. Whereas embedding theory tends to rely on methods inspiredby topological intuition, map theory depends more on group theory, especially onpermutation groups. Viewing a map as an incidence structure, especially when allfaces are triangles, leads naturally to designs and triple systems, as surveyed inChapter 13. It also leads to the pointline incidence structure of finite geometries,which is covered in Chapter 12.

An isomorphism of graphs is a bijection of the vertex-sets and of the edge-setsthat respects the incidence structure. A homeomorphism of graphs as topologicalspaces induces a graph isomorphism, but there are many homeomorphismsinducing the same graph isomorphism (just as there are many homeomorphismsof the unit interval taking 0 0 and 1 1). An isomorphism between a graphand itself is called an automorphism; the set of all automorphisms of a graph Gforms a group, denoted by Aut(G).

An isomorphism of graph embeddings G S and G S is an isomorphismof the underlying graphs that takes face boundaries to face boundaries. Ahomeomorphism from S to S taking the image of G to the image of G induces anisomorphism of the embeddings, but again there are many such homeomorphismsinducing the same isomorphism. An isomorphism between an embedding G Sand itself is called an automorphism, and the set of all such automorphisms is asubgroup of Aut(G).

Planarity and colouringThe development of topological graph theory as a distinct branch of graph theorywas motivated by two problems regarding planar graphs. The first problemis concerned with map-colouring. The chromatic number of a map is mostconveniently defined to be the chromatic number of the dual graph for that map that is, as the smallest number of colours needed to colour the dual vertices so thatdistinct endpoints of a dual edge get different colours. The chromatic number of asurface is the maximum of the set of numbers that occur as chromatic numbers ofmaps on that surface or, equivalently, as the maximum of the set of numbers thatoccur as chromatic numbers of graphs on that surface. The four-colour problemis to prove that every planar map has chromatic number at most 4. Its first knownwritten mention is in a letter from De Morgan to Hamilton in 1852. It was solvedby Appel and Haken [2] in 1976.

The second problem, called the planarity problem, is to characterize the graphsthat are planar. The solution by Kuratowski [20] characterizes them completely interms of two forbidden types of subgraphs that is, homeomorphic copies of thecomplete graph K5 and the complete bipartite graph K3,3. The generalization of


Kuratowskis theorem seeks, for each surface, a complete finite set of obstructionsto embeddability in that surface.

On the other hand, if a graph is not planar, rather than embed it in a higher genussurface, we may still want to draw it in the plane, yet now allowing edge-crossings.The crossing number of a graph G can be described intuitively as the minimumnumber of edge-crossings in a drawing of G in the plane; it is another measure ofthe extent to which a graph fails to be planar. After starting some decades ago withthe determination of a few difficult special cases, this pursuit has emerged into amore general topic, involving forbidden subgraphs, as described in Chapter 7.

4. Rotation systems

Besides finding a flaw in Kempes attempted proof of the four-colour theorem,Heawood [16] expanded the quest to finding the colouring numbers of all closedsurfaces. In 1890, he proved that the quantity

H() = 12(7 + 49 24 ) which is now called the Heawood number of a surface of Euler characteristic is an upper bound for the chromatic number of the surface. Proving that itcannot be improved for any surface except the Klein bottle N2 became knownas the Heawood problem. Its solution, completed in 1968 by Ringel and Youngs[30], required the construction of minimum-genus embeddings for the completegraphs. The construction employs a combinatorial method for specifying thoseembeddings that originated with Heffter.

In an embedding in an oriented surface, the rotation at a vertex is the cyclicordering of the edge-ends incident to that vertex, induced by the specifiedorientation. The set of all rotations is called the rotation system (or rotationscheme). Detailed examples of rotation systems are given, for instance, inSection 3.2 of [13] and by Section 6.6 of [44].

It is not hard to see how to trace out the face boundaries of the embeddingusing only the rotation system. Thus, every rotation system on a graph that is, anassignment of a cyclic order to the edge-ends incident to each vertex determinesan oriented embedding. In his proof that the Heawood bound is achievable for theorientable surfaces Sh with h = 1, 2, . . . , 6, Heffter [17] listed the vertices on theboundary walk of each face. This form of embedding specification was dualized byEdmonds [5] into the form more widely used and led to the Edmonds algorithmof determining the minimum genus of a graph by inspecting all its possible rotationsystems.


A rotation system can be encoded by two permutations of the set of directededges: a permutation whose cycles are the rotation of the directed edgesbeginning at each vertex, and an involution that takes each directed edge toits reverse. To trace the face boundaries, one merely computes the cycles of .The permutation group generated by and (with the specification of these twogenerators) is called the monodromy group or dart group of the map. Thus, wemight view an oriented embedding as no more than a permutation group with twospecified generators, one of which is an involution without fixed points. Althoughthis viewpoint removes geometrical intuition, it is particularly helpful for computerconstruction, for instance, of all highly symmetric maps of low genus (see [4]).

Viewing an embedding as a permutation group on the set of directed edgesintroduces terminology, notation and techniques from the theory of groups actingon sets. A (left) action of the group A on the set W is a homomorphism from Ainto the group of permutations on the set W , where we write permutations on theleft aw (prefix notation). If the homomorphism is injective, the action is said tobe faithful. For a faithful action, if aw = w for all w W , then a is the identity.The stabilizer of a W , denoted by Aw, is the subgroup of all a A such thataw = w. An action of A on W is transitive if, for all w, z W there is an a Asuch that aw = z. A transitive action is regular if the stabilizer Aw is trivial: it iseasy to show that all stabilizers for a transitive action are conjugate inA, so if oneis trivial, they are all trivial.

Given an oriented embedding, an automorphism of the graph is anautomorphism of the embedding if and only if the graph automorphism eitherpreserves the rotation at every vertex, or reverses the rotation at every vertex. Inthe former case, the automorphism is orientation-preserving and in the latter caseit is orientation-reversing.

Rotation systems for non-orientable embeddings are more complicated. Firstwe choose one of the two possible cyclic orderings of the edges incident to eachvertex induced by the embedding. Then each edge is assigned one of two possibletypes, flat or twisted (alternatively called type-0 and type-1, or sign + and sign ),depending on whether or not an open neighbourhood of the edge can be given anorientation consistent with the rotations at its endpoints. It is not hard to use theinformation of vertex rotations and edge types to trace the face boundary walks, soany such assignment of a general rotation system to a graph defines an embeddingof the graph. In this case, however, apparently different general rotation systemscan define the same embedding: we can always choose to reverse the cyclic orderat a vertex in exchange for reversing the type of all edges incident to the vertex.

Two rotation systems are equivalent if we can get from one to the other by asequence of such moves. Notice that a general rotation system for an orientableembedding may have twisted edges.Ageneral rotation system defines an orientableembedding if and only if each cycle contains an even number of twisted edges or,


equivalently, if and only if there is an equivalent general rotation system for whichall edges are flat.

Automorphisms of embeddings can be interpreted in terms of general rotationsystems as follows. Given a general rotation system for a graph G, a graphautomorphism f of G gives an automorphism of the associated embedding if andonly if there is an equivalent general rotation system such that f either preservesall vertex rotations or reverses all vertex rotations and f preserves edge types.

Coding general embeddings as permutation groups uses ideas of Tutte [41].Given a map, add a vertex at the midpoint of each edge and at the centre of eachface, and then subdivide the map into triangles by adding edges from each originalvertex to each new vertex in an incident face and from each new edge-vertex toeach of the two new vertices in the faces incident to that edge, as indicated inFig. 3. (The process constructs the first barycentric subdivision of a 2-complex.)

Fig. 3. Using flags to encode an embedding

The resulting triangles are called flags and the embedding is determined whenwe specify three involutions on the set of flags, telling us which pairs of sidesof flags to identify: T for vertex-edge sides, L for edge-face sides, and R forvertex-face sides. Since each edge lies on a diamond of four flags identifiedalternately along vertex-edge sides and edge-face sides, the permutation LTLTis the identity. Thus, we can view a map as a permutation group, called againthe monodromy group, generated by three fixed-point-free involutions T ,L,Rsatisfying LTLT = 1. A map given by such a permutation group is orientable ifand only if the subgoup of the monodromy group consisting of even length wordsin T ,L,R has index 2 (it necessarily has index at most 2). This abstract group-theoretic view of maps proves useful in computer constructions. The idea of a mapas a group is developed in Chapter 11, with particular attention to maps supportingregular actions by automorphisms on directed edges, flags or vertices.


5. Covering spaces and voltage graphsIn Ringels initial work on the Heawood problem in the early 1950s, he usedrotation systems and surgery that is, modifying an embedding by cutting apartthe embedding surface and sewing on a handle or crosscap to allow the additionof extra edges. Surgery was a well-established technique by the early 20th century(see [32]).

Ringels innovation was to amplify the power of rotation systems by usingone, two or three rotations algebraically to generate the remainder of thesystem. He masterfully designed generating rotations that would solve intricatespecial problems. After Gustin [15] created a remarkable computational aid (nowcalled a combinatorial current graph) to construct generating rotations, Youngsjoined Ringel in a five-year pursuit of the complete solution, which involvedmany different forms of combinatorial current graph, each defined by differentrules. The complete solution, announced in [30], occupied about 300 journalpages.

Topologically, a covering (or covering projection) p: X X of surfacesor graphs is a continuous surjection satisfying this condition: each point x ofthe codomain X has a neighbourhood U such that every component of p1(U)is mapped homeomorphically onto U . For a surface, if this condition holdseverywhere except for a finite number of points of the codomain, then the mappingis called a branched covering; its topological abstraction from Riemann surfacescan be traced back to Alexander [1] and A. W. Tucker [37].

Of particular interest are the regular coverings, which are obtained from theaction of a groupA of automorphisms on X, where X = X/A, the quotient spaceof orbits of A, and where p: X X/A is the natural quotient projection. Forgraphs we need the action to be fixed-point-free to get a covering, and for surfaceswe need the action to be fixed-point-free except at a finite number of points to geta branched covering. For other group actions, the orbit space has the structure ofan orbifold, a concept that plays a key role in Thurstons study of geometricalstructures on 3-manifolds (see [36]).

Whereas the topological theory of covering spaces describes an existentialrelationship between the domain and the codomain of a mapping, the theoryof voltage graphs, due to Gross [7] and Gross and Tucker [12], provides acombinatorial tool for constructing graphs and graph embeddings. In voltage graphtheory, the many specialized forms of combinatorial current graph originating withGustin and augmented by Ringel and Youngs (see [29]) are all unified, so thatthe RingelYoungs embeddings are readily understood as the duals of coveringsof voltage graphs (see [9] and [11]). Moreover, the power of the technique wasamplified so that it applies not only to embeddings of complete graphs, but to anyembedding with sufficient symmetry.


One view of a covering is that every cycle in the base graph unwinds when liftedto the covering graph. It is natural, therefore, to look at coverings where cyclesunwind completely, just as the unit circle unwinds to the real line when lifted viathe covering eit . Thus, every finite graph that is not a tree is covered by an infinitetree, where vertices repeated by a walk in the graph may now become separatewhen the walk is lifted to the tree. Coverings lead inevitably to infinite graphs,which have a far more complicated topological structure, whether embedded ornot. Chapter 14 gives an introduction to the topology of infinite graphs.

Regular voltages

Given a digraph G = (V ,E), a regular voltage assignment in a group B is afunction :E B that labels each edge e with a value (e): the pair G, is called a regular voltage graph; the graph G is called the base graph; the group B is called the voltage group; the label (e) is called the voltage on the edge e.It is said to specify the covering digraph G , defined as follows:

V (G) = V = V B, the Cartesian product; E(G) = E = E B; if the edge e is directed from a vertex u to a vertex v in G, then the edge

eb = (e, b) in G is directed from the vertex ub = (u, b) to the vertex vb(e) =(v, b(e)).

A more general concept, called a permutation voltage graph, was introducedin [12]. Whereas every regular covering, in the sense of topology, is realizable bya regular voltage assignment, all coverings (including the non-regular coverings)are realizable by permutation voltage assignments.

Vertices and edges of the covering graph are usually specified in a subscriptednotation, rather than in Cartesian product notation. There is a standard exception tothis convention, intended to avoid double subscripting. The digraph G is usuallycalled simply the covering graph. Moreover, its underlying (undirected) graphis also denoted by G and is also called the covering graph. Using such sharedterminology avoids excessively formal prose; in context, no ambiguity results.Fig. 4(a) shows a regular voltage graph G,:E Z3, and Fig. 4(b) shows thecorresponding covering graph.

Assigning an involution x as the voltage to a loop e at a vertex v in the basegraph causes the e-edges in the covering digraph to be paired that is, the directededge eb from a vertex vb to a vertex vb+x is paired with the directed edge eb+xfrom vb+x to vb. The term covering graph also refers to the undirected graph that is


u0

u1

u2

v0

v1

v2

b1u v

b

c

0

11

a

voltages in Z3

c2c1

c0

a2

a1

b2

b0

a0

(a) (b)

Fig. 4. A regular voltage assignment and the covering graph

obtained by identifying these pairs of directed edges as a single edge, even thoughthat usage conflicts with standard topological usage. (The present authors usedderived graph in their earlier papers to avoid this discrepancy in usage.)

Let G = (V ,E), :E B be a regular voltage graph. The graph mappingfrom the covering graph G to the voltage graph G, given by the vertex functionand edge function

vb v and eb erespectively, is called the natural projection. (Thus, the natural projection is givenby erasure of subscripts.) Let G be the covering graph for a regular voltagegraph G = (V ,E), :E B. Then the vertex subset {v} B = {vb: b B} is called the (vertex) fibre over v; similarly, the edge subset {e} B = {eb: b B} is called the (edge) fibre over e.

An assignment of voltages to a graph also induces an assignment of voltages todirected walks, simply by taking the product of the voltages. Many of the propertiesof the covering graph for example, whether it is connected can be stated interms of the voltages assigned to closed walks, all starting from a root vertex.

If the base graph is a bouquet and the set X of voltages appearing on edgesgenerates the voltage group A, then the covering graph is a Cayley graph for thegroup A, denoted C(A, X). The usual definition of C(A, X) is the graph withvertex-set A and a directed edge labelled x from a to ax, for every a A andevery x X; left multiplication by A provides the action of A on C(A, X) byautomorphisms. The graph underlying C(A, X), without directions and labels andwith parallel edges identified, is also called a Cayley graph and is denotedC(A, X).

White [43] defined the genus of a group as the minimum genus taken over allits Cayley graphs. Proulx [25] classified the toroidal groups, which fall into 17infinite families plus some sporadic cases, and Tucker [38] established that thereare finitely many groups of each genus greater than 1 and only one of genus 2.


The faces of an embedding of a Cayley graph C(A, X) in a surface providerelators: words in the generators that reduce to the identity. The hope of Burnside,Dehn and others, when they first studied embeddings of Cayley graphs 100 yearsago, was that these geometrical pictures of a group might reveal the algebraicstructure of a group via a presentation in terms of generators and relators. Such apresentation is given in a format like this:

x, y, z: x2 = y2 = z2 = 1, (xy)2 = (yz)3 = (xz)4 = 1.There are various logical difficulties with such presentations, the most famous one(established by Michael Rabin [26]) being the non-existence of an algorithm todecide whether any given presentation defines the trivial group. On the other hand,a group presentation is an extremely efficient way of describing a group and is thebasis for all computer calculations for groups (and therefore maps).

Lifting embeddingsWhen a voltage graph is cellularly embedded in a surface, we obtain an embeddingfor the covering graph by using the rotation system for the base embedding to defineone for the covering graph: vertex rotations and edge types are the same as thosein the base graph with subscripts erased. Moreover, the graph projection extendsto a branched covering from the covering embedding to the base embedding, withbranching inside any face whose boundary walk has non-identity voltage. (See [8]for a more complete description and illustrative examples.)

When the base graph is a bouquet and the voltages X generate the voltage groupA, the resulting embedding is called a Cayley map (see [3] and [28]), which can bedenoted by CM(A, ), where is given as a single cycle of the elements of X andtheir inverses. For example, CM(Z7, (1, 3, 2,1,3,2)) defines a triangularembedding of K7 in the torus having a fixed-point free action of Z7. The minimumgenus over all Cayley maps for a group, which is called the strong symmetricgenus, is always at least as large as the genus (see Tucker [39]) and is easier tocompute in many cases. A survey of the genus and other group parameters like thestrong symmetric genus is given in Chapter 10.

6. EnumerationEnumeration is a classical pursuit in mathematics, and the development of powerfulcounting methods for graphs preceded their adaption to counting topologicalobjects. Calculating surface-by-surface inventories of embeddings of a givengraph, programatically initiated by Gross and Furst [10], combines some of theprincipal methods of embedding construction, especially rotations and surgery,


with a variety of standard enumerative methods. Such inventories are the topic ofChapter 3.

In recent years, Kwak and Lee have led in the application of voltage graphmethods for enumerating graph coverings, and Chapter 9 provides an account ofthis active branch of topological graph theory.

Combinatorial methods predominated in the older, complementary programmeof research launched by Tutte [40], [41] into the counting of maps on a givensurface. Jackson and Visentin [19] have provided a complete listing of the mapswith a small number of edges.

7. Algorithms

The prototypical embeddability problem in topological graph theory is to determinea formula for the minimum genus of the graphs in an infinite class. The outstandingsingle example is the RingelYoungs formula

min(Kn) = 1

12 (n 3)(n 4)

for the genus of a complete graph. Ringel also derived formulas for the minimumgenus of hypercube graphs and of complete bipartite graphs.

With such success, there arose the question of the existence of a polynomial-timealgorithm to calculate the minimum genus. Planarity testing may have seemed aninitial step toward the more general goal. There were informal methods that ratherquickly made the planarity decision, and some formal quadratic-time algorithmswere developed. The naive algorithm based on this Kuratowski characterizationdoes not run in polynomial time. Nonetheless, when iterative application ofthe Jordan curve theorem is included in the test, a quadratic-time algorithm isachievable. Ultimately, Hopcroft and Tarjan [18] produced a linear-time algorithm.

Eventually, Thomassen [35] showed that deciding whether a given graphhas a given minimum genus is an NP-complete problem. Interestingly, Mohar[21] subsequently showed that, for each fixed surface, the problem of decidingthe embeddability of a given graph is solvable in linear time; however, themultiplicative constant grows rapidly with increasing surface genus. Chapter 4surveys algorithms for embeddings.

Interest in calculating the maximum genus began with Nordhaus, Stewart andWhite [23]. Although the obvious algorithm based on Xuongs characterization[45] requires exponential time, Furst, Gross and McGeoch [6] derived apolynomial-time algorithm. Chapter 2 surveys the main results on the maximumgenus of a graph.


8. Graph minorsThere are two natural ways to reduce graphs for the purposes of inductive proofs:deleting an edge (and any isolated vertices that this creates) or contracting an edge,which is defined in a combinatorial context as identifying its endpoints and deletingall resulting loops and multiple edges.Aminor of a graphG is any graph obtainablefrom G by a sequence of edge deletions and edge contractions. Clearly, if G canbe embedded in the surface S, then so can each of its minors. Thus, if G is notembeddable in S, it has a minimal minor that is not embeddable in S, a forbiddenminor. As Wagner [42] first observed, Kuratowskis theorem immediately impliesthat the forbidden minors for the plane are K5 and K3,3.

One of the principal objectives of topological graph theorists for about fifty yearswas to prove a Kuratowski-type theorem for non-planar surfaces S, that the set offorbidden minors for embeddability in S is finite. This goal was finally attained byRobertson and Seymour [31], and this is described in Chapter 5. Furthermore, theybootstrapped from their results for surfaces to a proof of Wagners conjecture thatunder the partial ordering on all graphs by the minor relation, there are no infiniteantichains: in any infinite collection of graphs G1,G2, . . . , there exist indices iand j with i < j , such that Gi is a minor of Gj . Their series of twenty papers ongraph minors over the last twenty years is a landmark in graph theory, providingan overarching structure for all graphs.

Along with the theory of minors, the concepts of edge-width, face-width,and representability have evolved as powerful tools for the understanding ofembeddings. The underlying theme is that graphs that are 3-connected and locallyplanar (that is, every vertex is contained in a large planar subgraph) should behavelike planar graphs. Chapter 6 applies this theme to colouring embedded graphs.

References1. J. W. Alexander, Note on Riemann spaces, Bull. Amer. Math. Soc. 26 (1920), 370372.2. K. Appel and W. Haken, Every planar map is four-colorable, Bull. Amer. Math. Soc. 82

(1976), 771772.3. N. L. Biggs and A. T. White, Permutation Groups and Combinatorial Structures,

Cambridge University Press, 1979.4. M. Conder and P. Dobcsnyi, Determination of all regular maps of small genus,

J. Combin. Theory (B) 81 (2001), 224242.5. J. R. Edmonds, A combinatorial representation for polyhedral surfaces, Abstract in

Notices Amer. Math. Soc. 7 (1960), 646.6. M. Furst, J. L. Gross and L. McGeoch, Finding a maximum-genus graph imbedding,

J. Assoc. Comp. Mach. 35 (1988), 523534.7. J. L. Gross, Voltage graphs, Discrete Math. 9 (1974), 239246.8. J. L. Gross, Voltage Graphs, Section 7.4 of Handbook of Graph Theory, CRC Press,

2004.


9. J. L. Gross and S. R.Alpert, The topological theory of current graphs, J. Combin. Theory(B) 17 (1974), 218233.

10. J. L. Gross and M. L. Furst, Hierarchy for imbedding-distribution invariants of a graph,J. Graph Theory 11 (1987), 205220.

11. J. L. Gross and T. W. Tucker, Quotients of complete graphs: Revisiting the Heawoodmap-coloring problem, Pacic J. Math. 55 (1974), 391402.

12. J. L. Gross and T. W. Tucker, Generating all graph coverings by permutation voltageassignments, Discrete Math. 18 (1977), 273283.

13. J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley, 1987, and Dover, 2001.14. J. L. Gross and J. Yellen, Handbook of Graph Theory, CRC Press, 2004.15. W. Gustin, Orientable embedding of Cayley graphs, Bull. Amer. Math. Soc. 69 (1963),

272275.16. P. J. Heawood, Map-colour theorem, Quart. J. Math. 24 (1890), 332338.17. L. Heffter, ber das Problem der Nachbargebiete, Math. Annalen 38 (1891), 477580.18. J. Hopcroft and R. E. Tarjan, Efficient planarity testing, J. Assoc. Comp. Mach. 21

(1974), 549568.19. D. M. Jackson and T. I. Visentin, An Atlas of the Smaller Maps in Orientable and

Nonorientable Surfaces, Chapman & Hall/CRC, 2001.20. K. Kuratowski, Sur le problme des courbes gauches en topologie, Fund. Math. 15

(1930), 271283.21. B. Mohar, A linear time algorithm for embedding graphs in an arbitrary surface, SIAM

J. Discrete Math. 12 (1999), 626.22. B. Mohar and C. Thomassen, Graphs on Surfaces, Johns Hopkins Univ. Press, 2001.23. E. A. Nordhaus, B. M. Stewart and A. T. White, On the maximum genus of a graph,

J. Combin. Theory (B) 11 (1971), 258267.24. C. D. Papakyriakopoulos, A new proof of the invariance of the homology groups of a

complex, Bull. Soc. Math. Grce 22 (1943), 1154.25. V. K. Proulx, Classification of the toroidal groups, J. Graph Theory 2 (1981), 269273.26. M. O. Rabin, Recursive unsolvability of group theoretic problems, Ann. of Math. (2)

67 (1958), 172194.27. T. Rado, ber den Begriff der Riemannschen Flache, Acta Litt. Sci. Szeged 2 (1925),

101121.28. B. Richter, J. irn, R. Jajcay, T. Tucker and M. Watkins, Cayley maps,

J. Combin. Theory (B) 95 (2005), 489545.29. G. Ringel, Map Color Theorem, Springer, 1974.30. G. Ringel and J. W. T. Youngs, Solution of the Heawood map-coloring problem,

Proc. Nat. Acad. Sci. USA 60 (1968), 438445.31. N. Robertson and P. Seymour, Graph minors XX. Wagners Conjecture,

J. Combin. Theory (B) 92 (2004), 325357.32. H. Seifert and W. Threllfall, Lehrbuch der Topologie, Chelsea, 1947; English transl. by

J. Birman and J. Eisner, Academic Press, 1980.33. P. D. Seymour, Sums of circuits, Graph Theory and Related Topics (Proc. Conf. Univ.

Waterloo, 1977), Academic Press (1979), 341355.34. G. Szekeres, Polyhedral decompositions of cubic graphs, Bull. Austral. Math. Soc. 8

(1973), 367387.35. C. Thomassen, The graph genus problem is NP-complete, J. Algorithms 10 (1989),

568576.


36. W. P. Thurston, Three-Dimensional Topology and Geometry, Princeton Univ. Press,1997.

37. A. W. Tucker, Branched and folded coverings, Bull. Amer. Math. Soc. 42 (1936),859862.

38. T. W. Tucker, The number of groups of a given genus, Trans. Amer. Math. Soc. 258(1980), 167179.

39. T. W. Tucker, Groups acting on surfaces and the genus of a group, J. Combin. Theory(B) 34 (1983), 8298.

40. W. T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 2138.41. W. T. Tutte, A census of planar maps, Canad. J. Math. 15 (1963), 249271.42. K. Wagner, ber eine Eigenschaft der ebene Komplexe, Math. Ann. 114 (1937),

570590.43. A. T. White, On the genus of a given group, Trans. Amer. Math. Soc. 173 (1972),

203214.44. A. T. White, Graphs of Groups on Surfaces: Interactions and Models, North-Holland,

2001.45. N. H. Xuong, How to determine the maximum genus of a graph, J. Combin. Theory (B)

26 (1979), 217225.

2Maximum genus

JIANER CHEN and YUANQIU HUANG

1. Introduction2. Characterizations and complexity3. Kuratowski-type theorems4. Upper-embeddability5. Lower boundsReferences

Since the introductory investigation by Nordhaus, Stewart and White, themaximum genus of a graph has attracted considerable attention frommathematicians and computer scientists. In this survey, we focus on theprogress in recent years. In particular, we study its characterizations,algorithmic complexity, upper-embeddability and lower bounds.

1. Introduction

The maximum genus max(G) of a graph G is the greatest integer k for whichthere exists a cellular embedding of G into the orientable surface of genus k. Forexample, the maximum genus of a tree or a cycle is 0, and the maximum genus ofthe complete graph K4 is 1.

By Eulers polyhedron formula, if a cellular embedding of a graph G with nvertices, m edges and r faces is on a surface of genus , then

n m + r = 2 2.

Since r 1, we have 12 (m n + 1). The number (G) = m n + 1 iscalled the cycle rank of G. It follows that the maximum genus of G is boundedabove by 12(G).

34

2 Maximum genus 35

The maximum genus is additive over 2-edge-connected components, in thesense that if e is a cut-edge of a connected graph G and if G1 and G2 are thecomponents of G e, then max(G) = max(G1) + max(G2) (see [19]). Wetherefore need to consider only 2-edge-connected graphs.

An ear decomposition D = [P1, P2, , Pr ] of a graph G is a partition of theedge-set of G into an ordered collection of edge-disjoint paths P1, P2, , Pr suchthat P1 is a cycle and, for i 2, Pi is a path with only its endpoints in commonwith P1 Pi1. Each path Pi is called an ear. It is well known (see [24]) thata graph G has an ear decomposition if and only if it is 2-edge-connected.

The operations of edge-insertion and edge-deletion have turned out to be usefulin the study of graph embeddings. Let (G) be an embedding of a graph G. We saythat a new edge e is inserted into (G) if the two endpoints of e are inserted intothe corners of faces in (G), yielding an embedding of the graph G+ e. If the twoendpoints of e are inserted into corners of the same face f in (G), then the edgee splits the face f into two faces and leaves the embedding genus unchanged.In this case, the two sides of the new edge e belong to two different faces in theresulting embedding of G + e (see Fig. 1(a)).

(b)(a)Fig. 1. Inserting an edge

On the other hand, if the two endpoints of e are inserted into corners of twodifferent faces f1 and f2 in (G), then the edge e merges the faces f1 and f2 intoa single larger face and this increases the embedding genus by 1. In this case, thetwo sides of the new edge e belong to the same face in the resulting embedding ofG+e. Topologically, this operation is implemented by cutting along the boundariesof the two faces f1 and f2, leaving two holes on the surface, and then adding ahandle to the surface by pasting the two endpoints of an open cylinder to theboundaries of the two holes, so that the new edge e now runs along the new handle(see Fig. 1(b)).

The discussion of the edge-deletion operation can be described in the reversedorder. Let (G) be an embedding of a graph G, and let e be an edge that isnot a cut-edge. If the two sides of e belong to two different faces in (G), thendeleting e from (G) merges the two faces without changing the embeddinggenus; if the two sides of e belong to the same face in (G), then deletinge from (G) splits the face into two faces and decreases the embeddinggenus by 1.

36 Jianer Chen and Yuanqiu Huang

2. Characterizations and complexityIt is an interesting problem to characterize the maximum genus of a graph in termsof the combinatorial structures of the graph. This leads to a better and deeperunderstanding of maximum genus, while a combinatorial characterization maylead to efficient constructions of maximum-genus embeddings. There have beenseveral successful characterizations of maximum genus, and this section gives asummary of them.

Let T be a spanning tree of a graph G. The edge complement GT is called aco-tree; note that the number of edges in any co-tree of G is the cycle rank (G).A co-tree G T need not be connected, and a connected component of G T iseven or odd, according to the parity of the number of edges in it.

The deficiency (G) of a graph G is the minimum number of odd componentsin any co-tree of G. Any tree whose co-tree achieves this minimum is called aXuong tree. The following result is due to Xuong [25]:

Theorem 2.1 For every graph G, max(G) = 12 ((G) (G)).

The proof of Theorem 2.1 is based on the observation that properly addingtwo adjacent edges e1 and e2 to an embedding (G) of a graph

052180230X_Graph

Documents

Transcript of 052180230X_Graph