Towards the Construction of a Fast Algorithm for the Vertex Separation Problem on Cactus Graphs...

Towards the Construction of a Fast

Algorithm for the Vertex Separation

Problem on Cactus Graphs

Minko Markov

Sofia University, Faculty of Mathematics and Informatics

[email protected]

Structure of the presentation Background Vertex Separation of Trees and

Unicyclics Vertex Separation of Cacti Boudaried Cacti and Stretchability Decomposition of Boundaried Cacti Main Theorem for Stretchability on

Boundaried Cacti

April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University

Vertex Separation (VS) of Layouts and Graphs An NP-complete problem on undirected

ordinary graphs Do not confuse “Vertex Separation” with

“Vertex Separator” The definition of Vertex Separation is

based on the definition of linear layout


vs(G) = min {vsL(G) | L is a layout of G} = 2

VS of Layouts and Graphs (2)

April 11, 2023

vsL(G)=2

0x

y

vu

w

u v w y x

1 2 2 2

L

G = (V,E)

πL(u) = {u}πL(v) = {u,v}πL(w) = {v,w}πL(y) = {w,y}πL(x) = ∅Minko Markov, Faculty of Mathematics and Informatics, Sofia University

(u,v), (u,w), (v,w), (v,y), and (w,y) are clean

Node Search Number (SN)

April 11, 2023Minko Markov, Faculty of Mathematics and Informatics. This research is supported

by Sofia University Science Fund under project "Discrete Structures"

w+

all edges are contaminated(u,v), (u,w) and (v,w) are clean

xy

vu

wu+ v+ x+y+ v—u— y— w—x—

sns(G) = 3

S =

(u,v) is cleanall edges are clean

monotonous (progressive) search

VS is equivalent to SN

For every graph G, vs(G) = sn(G) − 1 Optimal searches define unique optimal

layouts, optimal layouts define multitudes of optimal searches

April 11, 2023

x y

vu

w L = u v w y x, vsL(G) = 2

S = u+ v+ w+ u− y+ v− x+ y− x− w−, sns(G) = 3

Minko Markov, Faculty of Mathematics and Informatics, Sofia University

Fast algorithms for VS on restric-ted graphs

O(n) for trees (Ellis, Sudborough, Turner, 1994) O(n lg n) on unicyclic graphs (Ellis, Markov,

2004), improved to O(n) (Chou, Ko, Ho, Chen, 2006)

O(bc + c2 + n) on block graphs (Chou et al., 2008) O(n) on 3-Cycle-Disjoint Graphs—a strict

subclass of cactus graphs (Yang, Zhang, Cao 2010)


Cactus graphs (cacti)

April 11, 2023


Rooted Cacti


VS of trees – O(n) algorithm by Ellis, Sudborough, Turner (1994) Theorem (EST, 1994): If T is a tree and

k ≥ 1, then vs(T) ≤ k iff every vertex induces at most two subtrees of vs = k.

April 11, 2023

vvs = k vs = k

< k < k < k


k-critical subtree

T is a rooted tree, vs(T) = k, and the root induces two subtrees of vs = k.

April 11, 2023

k k< k < k...


Label of a tree

April 11, 2023

T2TT1

lab(T) = (k, p, q), k > p > q

p pk k

qvs(T)=kvs(T1)=pvs(T2)=q


The EST algorithm

April 11, 2023

▲ _ _ ▲ _ _ ▲ _ lab:

lab = ?

lab1 = (5,2)

lab1:9 8 7 6 5 4 3 2 1

lab2 = (7,6,5)

lab3 = (8,5,2c)

lab2:

lab3:

_ _ _ ▲ _ _ ● _ _ ▲ ▲ ● _ _ _ _

● _ _ _ _ _ _ _ _ ● _ _ _ _ _ _ _ ● _ _ _ _ _ _ ● _ _ _ _ _ _ ● _ _ ● _ _ ● _ _ _

lab = (9)


The VS backbone of a tree the easiest kind of rooted tree of VS k


VS < k VS < kVS < k

VS = k1

K−1 K−1 K−1

The VS backbone of a tree the second best kind (VS = k)


VS = kVS < kVS < k

VS = k1

K K−1 K−1

The VS backbone of a tree an even harder rooted tree of VS k


VS = k VS = kVS < k

VS = k1

K K−1 K

The VS backbone of a tree the hardest kind


VS = k VS = kVS < k

VS = k

1

K K−1 K

VS = k-1 VS = k-1

1

1

K−1 K−1

The backbone of a non-rooted tree


VS < k

VS = k1

K−1 K−1 K−1K−1 K−1 K−1 K−1 K−1 K−1

1 1

Vertex Separation of Cacti

Theorem (M.M., 2007). Let G be a cactus and k ≥ 1. Then vs(G) ≤ k iff:Every vertex induces at most two cacti of

separation k, all others are < k.In every cycle there exist vertices u and v

(not necessarily distinct) such that G [u,v]⊝ is k-stretchable.


G⊝[u,v]

April 11, 2023

u v

G G [u,v]⊝


Stretchability k w.r.t. u and v


u v

K

The idea behind the theorem Definition: a c-path (cactus path) in a cactus

is a linear order of vertices and cycles


a

b

e

g

c

kid

tq

rponm

lj

hf vu xs3

s2s1

w

s4

C = a s1 f g h s2 m n o s3 u v w x

The backbone of a cactus


bfd

he

ca is

j

K−1 K−1 K−1 K−1 K−1 K−1K−1K−1

K

The root and the backbone


r

bfd

he

ca is

jK

lab(G(r)) = ( K,

G

lab(G1(r)) )

G1

The root and the backbone


r

fd

he

s

lab(G(r)) = ( K,

i jb caK

G

lab(G1(r)) )

G1

ic

The cacti pitfall

April 11, 2023

k-1 k-1

k k

k k


The cacti pitfall

April 11, 2023

k k

k-2 k-2

k k


The cacti pitfall


The solution for cacti?

Take Stretchability w.r.t. k vertex pairs as the primary problem

Consider bounaried cacti, the boundary being the vertices w.r.t. which we stretch

The original problem reduces to this one – just take an empty boundary


Boundaried cactus

A cactus G in which some cycles s1, …, sn have two boundary vertices each. All boundary vertices are of degree 2.

Let the boundary pair in si be ‹ui, wi›. The search game on G is performed so that n searchers are placed on U = {u1, …, un} initially and at the end, each of W = {w1, …, wn} must have a searcher.

The boundary is ‹U, W›.


The Residual VS of a boundaried cactus Let G be a boundaried cactus with n

vertex pairs in the boundary. Let k be the stretchability of G w.r.t. the boundary. Then rvs(G) = k – n. We proved k – n > 0 always.

From now on we consider RVS of boundaried cacti. VS of cacti is a special case of RVS.


RVS of Boundaried Cacti

Theorem: Let G be a boundaried cactus, boundary ‹U, W›, and m ≥ 1. Then rvs(G) ≤ m iff:Every nonboundary vertex induces at most two

boundaried cacti of rvs m, all others are < m.In every cycle there are nonboundary vertices x

and y (not necessarily distinct) such that G is (k+1)-stretchable w.r.t. ‹U {x}, W {y}› or ‹U {y}, W {x}›.


How to prove k-stretchability It is more rigorous to use the VS

definition and terminology, not the NSN


L is k-stretchable iff the separation of any vertex is (k – the number of intervals it is in)

u rx

{u,v,w} : left, {x,y} : right

v wy

r is the rightmost neighbour of x and y

layout L

How to prove k-stretchability It is easier to modify L into an extended

layout L* and consider its VS


xu v wy

layout Llayout L*

u v w

L is k-stretchable iff L* has VS ≤ k

Proof of the theorem, part I Consider an optimal extended layout L*.

Consider the leftmost and rightmost nonboundary vertices a and z.


az

s3s2s1 s4

rvs(G) ≤ 5 → rvs(G1) ≤ 4, i.e. vs(G1) ≤ 4

G1

G

THE END


Towards the Construction of a Fast Algorithm for the Vertex Separation Problem on Cactus Graphs...

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