On Characterizing Graphs with Branchwidth at Most

RICE UNIVERSITY

ON CHARACTERIZING GRAPHS WITH

BRANCHWIDTH AT MOST FOUR

Kymb erly Dawn Riggins

A Thesis Submitted

in Partial Fulfillment of the

Requirements for the Degree

Master of Science

Approved Thesis Committee

Dr Nathaniel Dean Chairman

ciate Professor Asso

Computational and Applied Mathematics

Dr Richard A Tapia

Noah Harding Professor

Computational and Applied Mathematics

Dr Matthew DeVos

Visiting Assistan t Professor

Computational and Applied Mathematics

Houston Texas

April

ON CHARACTERIZING GRAPHS WITH

BRANCHWIDTH AT MOST FOUR

Kymberly Dawn Riggins

Abstract

There are several ways in which we can characterize classes of graphs One such

way of classifying graphs is by their branchwidth In working to characterize the

class of graphs with branchwidth at most four we have found a set of reductions

that reduces members of to the zero graph We have also computed several pla

rs of the obstruction set O for graphs with branchwidth at most four nar membe

This thesis will summarize previous results on branchwidth and reveal the previously

mentioned new results

Acknowledgments

I would like to thank my committee members My advisor Dr Nathaniel Dean for

his b elief and p ersistence inpushingmetodowhatheknew I could do I appreciate

your honesty and candor as well as your exp ectation that I p erform at a level ab ove

what I am used to

Dr Richard Tapia for his guidance and inspirational and thought provoking

comments Words can not express how much of a p ositive impact you have made

on my life

Dr Matthew Devos for his willingness to be on my committee and enthusiasm

while doing so

I also want to thank former graduate students now tenure track professors Dr

IllyaV Hicks and William C Christian Their constantnudging and fatherly concern

has b een noted and appreciated and has had a ma jor role in who Iamtoday

I would also like to acknowledge my parents grandparents greatgrandparents

and extended family for the sacrices they made so that I can be who I am Who I

am and what I will accomplish is due to your love guidance supp ort and investment

I am eternally grateful

Last but not least I would like to thank my Lord and Savior Jesus Christ for

of me so that I can p erform to completion the talents and gifts He has placed inside

the tasks He has assigned to me

This research work was supp orted by the following National Science Foundation

Fellowships Diversity Graduate Program for Science and Engineering and the GK

Fellows Program

Table of Contents

Abstract ii

Acknowledgments iii

List of Figures vii

Introduction

Preliminaries

Motivation

Branchwidth

Treewidth and Branchwidth Denitions

Tangles and Separations

Previous Branchwidth Results

Obstruction Sets

Obstruction Sets and Treewidth

Obstruction Sets for Bounded Branchwidth Graphs

Obstruction Sets for Other Classes of Graphs

Computing Obstruction Sets

Reductions

Reduction Rules for Graphs with Branchwidth at Most Three

Reduction Rules for Graphs with Branchwidth At Most Four

Memb ers of O

Planar Members of O

Results

Conclusion and Future Work

Analysis of the Results

Potential NonPlanar Members of O

Bibliography

List of Figures

A graph and a corresp onding tree decomp osition

K and a corresp onding branch decomp osition

Middle set example

A separation of the Petersen graph

A Tangle of order ofthe Cub e

An example Halin graph

All complete graphs are chordal graphs

Reduction Rules for Graphs with G

Reduction Rules r r r

Reduction Rules R nd r and r r a

Planar Obstruction

viii

Planar Obstruction

Do decahedron

Icosahedron

Triplex

Tangle of order for b oth Basket and Triplex

Triplex and a branch decomp osition of width

Basket and a branch decomp osition of width

Chapter

Intro duction

Life is full of discrete optimization problems waiting to b e solved on real life instances

that can b e mo deled on graphs There are those problems for which no solution exists

but fortunately there are some of lifes problems that do actually have solutions

Unfortunately for any given instance of the solvable problems not all of them can b e

solved in a reasonable amountof timeon it The question then b ecomes For which

instances can lifes really dicult yet solvable problems be solved in a reasonable

amountof time

A tree is one of those sp ecial types of graph whose structure makes it easier to

solve extremely dicult problems on If a graph can be easily decomp osed into

a tree like graph then a dicult graph problem can be solved on that graph in a

erty that can indicate how reasonable amount of time Branchwidth is a graph prop

close a graph is to b eing a tree The lower the branchwidth of a graph the closer the

graph is to a tree Thus characterizing which graphs have small branchwidth would

be advantageous to those who need to determine how long it would take to solve a

NPhard graph problem This pap er will summarize the work done in that area as

well as provide additional work that furthers this eld of interest

b ehind this research as well as denitions and Chapter includes the motivation

imp ortant concepts needed to pro ceed Chapter fo cuses on the graph prop erty

branchwidth Chapter covers obstruction sets and the use of obstruction sets to

characterize graph classes Chapter introduces reductions that can be used to

reduce a graph with branchwidth at most four down to the empty graph Chapter

introduces members of the obstruction set of graphs with branchwidth at most four

Chapter isthe conclusion and includes future work and plans

Preliminaries

We consider a graph G V GEG V E to b e a collection of n no des and m

edges where n is the order and m is the size of the graph Eachedgee E G can

be represented as an unordered pair e u v In this case u and v are considered

to be adjacent to each other and e is incident with b oth u and v A graph G is

simple if no no de is adjacent to itself and at most one edge has the same two ends

A graph is connected if and only if it has a path between every pair of no des If

there exists two no des in a graph for which there is no path between them than

that graph is disconnected A graph is k connected if at least k no des must be

removed to disconnect the graph A graph is k edgeconnected if at least k edges

of order n has myst b e removed to disconnect the graph The complete graph K

every pair of distinct no des adjacent to one another A subgraph G V E of

G V E is a graph for which V V and E E A clique is a complete graph

that is a prop er subgraph of a graph

The class of graph problems for which a solution to each problem can be veried

eciently ie in p olynomial time is nondeterministic p olynomial time solvable

Problems in this class are often referred to as problems in NP If an algorithm is

olynomial time than known that actually nds the solution to a NP problem in p

that particular problem is considered to be deterministic p olynomial time solvable

these problems belong to a subset of NP called P NPhard problems are those

problems which are lab eled as inherently dicult If a problem is NPhard but is

also in NP than that problem b elongs to the NPcomplete set

Motivation

Fortunately many graph problems which are NPhard on general graphs can b e solved

in p olynomial and even linear time on graphs with b ounded branchwidth Even

though determining the branchwidth of a arbitrary graph has b een proven to b e NP

hard we do know that for xed k determining whether or not a graph has

branchwidth at most k is p olynomial Unfortunately the pro of of that result

do es not provide a way to construct such an algorithm but it merely proves existence

of one

From the work of Bo dlander and Thilikos we know that an eective decidable

algorithm to determine membership in the class of graphs with branchwidth at most k

can b e constructed and solved in linear time This algorithm is highly exp onential

in a p olynomial in k and thus is quite impractical for actual implementation and use

Being as such other ways are needed to characterize graphs with branchwidth at

most k

Although there are several dierent ways to characterize a graph class we have

chosen to fo cus our attention on the concept of using obstruction sets to characterize

graph classes We have chosen obstruction sets because when they are nite one can

test in p olynomial time if a graph G has a minor that is isomorphic to an elementin

the obstruction set

It is imp ortant to note that this metho d would be unrealistic to use if the ob

struction set was a set of innite size Fortunately a result that works in our favor is

the pro of of Wagners conjecture by Rob ertson and Seymour They proved that for

an innite graph class closed under minor taking wecancharacterize it with a set of

nite forbidden minor minimal subgraphs These forbidden minor minimal sub

graphs are called obstructions and the set comprised of them is called the obstruction

set for that class If a graph class F is closed under minors than its obstruction set

O is nite and a simple p olynomial algorithm that tests whether any member of

O is a minor of G and this determines if G F Minor testing can be done in

O n time and even O n time for sp ecial typ es of graphs The question we

now ask is how can one determine the obstruction set ofagraph class

Chapter

Branchwidth

Treewidth and Branchwidth Denitions

Before we lo ok at branchwidth we must rst lo ok at treewidth a closely related

graph prop erty whose denition preceded branchwidth Branchwidth has often been

used in conjunction with treewidth Tree decomp ositions and treewidth were rst

introduced by Rob ertson and Seymour in the second installment of their series on

Graph Minors A tree decomp osition of a graph G V GEG is a pair

a tree and X is a collection of jV T j subsets of V G where each T X where T is

no de of T is uniquely related to a memberofa X The following prop erties hold for

a tree decomp osition T X of a graph G

X V G

X V T

for all edges v w E G there exists an X V T with v X

i i

and w X

for all X X X V T if X is on the path from X to X in T

i j k j i k

then X X X

i k j

osition is one less than the maximum cardinalityof the The width of a tree decomp

X sets of X The treewidth of a graph G G is the minimum width over all

tree decomp ositions of G Figure is an example of a tree decomp osition of K

Branch decomp osition and branchwidth were intro duced shortly after treewidth

was introduced Given a graph G V E a branch decomp osition of G is a

pair T where T is a ternary tree and is a bijection from the leaves of T to the

edges of G A ternary tree is a tree with no des of degree either one or three

c d j h i a k e f g i,j

a,b,c c,d,e d,e,f d,f,h h,i

i,k

f,g

Figure A graph and a corresp onding tree decomp osition

31 5 1

4 2 56 e

2 4 6

Figure K and a corresp onding branch decomp osition

A relaxed branch decomp osition is a branch decomp osition with no des of degree

other than one and three

For a branch decomp osition T the middle set M e of an edge e E T

is comprised of no des v V G such that there are leaves t and t in dierent

comp onents of T V T ET e yet t and t are incident to v in G The

T is the number of nodes v V G b elonging to the middle order of an edge e E

set of e In other words the order of an edge is the cardinalityof M e For example

the order of e in Figure is four The width of a branch decomp osition is the

maximum order of all of the edges in a graph The branchwidth G of G is the

minimum width over all branch decomp ositions of G

Although branchwidth is not as popular a concept as treewidth it can reveal

information ab out a graph that the treewidth would not For edge centered graph

problems like the Traveling Salesman Problem the branchwidth of a graph reveals

31 5 a 1b

4 2 56 e

c d 3 2 4 6

M(e)={a,b,c,d}

Figure Middle set example

much more ab out the graph then the treewidth would The same go es for treewidth

and vertex centered graph problems like the Minimum Vertex Cover Problem where

the relationship between the edges of a graph are not of greater interest than that

of the no des of the graph Although they are dierent they are often used inter

changeably in terms of complexity results This is b ecause the treewidth of a graph

is b ounded ab ove and b elowby its branchwidth

G G b Gc

Tangles and Separations

tro duced by Rob ertson and Seymour in the tenth installmentof their Branchwidth in

Graph Minors pap ers is a closely related variant to the tangle number of a graph

A separation A B of a graph G is comprised of two distinct subgraphs A and

B of G such that

A B G

E A E B

The order of a separation is the cardinalityof the set V A V B

A tangle F in G of order is a collection of separations of order such that

the following holds

0 '' '' ' ' ' ' ' o:~/,~ ~~~;:-_-\,:,~o cf------6

A B

Figure A separation of the Petersen graph

for every separation A B of G of order one of A B orB A

is in F

S S

if A B A B A B F then A A A G and

if A B F then V A V G

For all graphs the tangle number is a lower b ound of the branchwidth of the graph

In certain situations they are equal If E G then

Gmax G

Previous Branchwidth Results

hwidth of a general graph G is NPhard Yet for some interesting Computing the branc

classes of graphs we can compute the branchwidth of the graphs in that class in

p olynomial time

Rob ertson and Seymour proved that the class of graphs with bounded branch

width is closed under taking minors They also showed that ngrid graphs have

branchwidth n In that same pap er Graph Minors X Rob ertson and Seymour

were able to characterize graphs with branchwidth at most k for k

A graph Ghas branc hwidth at most

Separation of order

Separations of order

v G v V G

Separations of order

fv w g G v w V G

e G n e e E G

Separations of order

fv w ug G v w u V G

such that v is not incident to e fv eg G n e e E Gand v V G

fe f g G n fe f g pairs e f E Gsuch that e and f share a no de

f g G n a

f g G n b

f g G n c

f g G n e

f g G n f

f g G n d

f g G n g

f g G n h

Figure A Tangle of order of the Cub e

if and only if each connected comp onentcontains at most one edge

if and only if each connected comp onent contains at most one vertex

of degree at most two

ifandonly if G do es not have K as a minor

Based on those results we know that factors have branchwidth star trees

trees with at most one nonleaf no de have branchwidth one and trees and forests

have branchwidth at most two Since outerplanar graphs are characterized as graphs

containing no K or K as minors then it is obvious to see that outerplanar graphs

have branchwidth at most two Series parallel graphs are also characterized as having

no K minor thus they have branchwidth at most two In addition to the results by

Rob ertson and Seymour Bo dlaneder was able to determine the obstruction set for

graphs with branchwidth at most three More information on that discovery will b e

presented in the next chapter

Even though only graphs with branchwidth at most three havebeencharacterized

there are still other results that allowus to characterize sp ecial classes of graphs with

b ounded branchwidth

Hicks was able to determine the branchwidth of Halin and chordal graphs as well

as develop aclass of graphs which provide an upp er b ound for branchwidth

Halin graphs are dened in terms of how they are constructed A Halin graph is

constructed from a planar emb edding of a tree with no degree vertices by adding a

cycle through the leaves of the tree according to the cyclic order dened by its planar

emb edding Halin graphs have branchwidth three

Figure An example Halin graph

Achordal graph is a graph for whichevery cycle of length at least four has an edge

between two nonconsecutive no des in the cycle The clique numb er of a graph G

is the size of the largest clique complete subgraph of the graph The branchwidth

of chordal graphs with at least no des is b ounded ab ove and below by the clique

number of the graph

d Ge G G

The H graphs are graphs that are constructed from complete bipartite graphs

des pro duces a graph Removing a onefactor from a complete bipartite graph on k no

which is lab eled M Let A B be the bipartition of the no de set of the bipartite

graph M Add edges to M so that the subgraph induced by A is complete The

k k

resulting graph is denoted as H Hicks showed that H is chordal and H k

k k k

Figure All complete graphs are chordal graphs

Figure H

Kloks et al showed that for split graphs and bipartite graphs the branchwidth

problem is NPcomplete For interval graphs it has b een proved that you there is a

O n logn algorithm to nd the branchwidth citeterr

Although these results are helpful they only reveal information ab out a small

subset of graphs More results are needed to be able to characterize graphs by their branchwidth

Chapter

Obstruction Sets

Let G V GEG and H V H EH be simple and connected graphs

The graph H is a minor of G if H can be obtained from G by a series of vertex

deletions edge deletions andor edge contractions A minor H of a graph G is

minor minimal with resp ect to a graph prop erty P if any additional minor

erty op eration on H pro duces a graph H which no longer p ossesses the graph prop

P that G p ossessed A graph class G is closed under minor taking when for any

graph G G if H a minor of G then H G Note If H is a minor of a G then

H G

In their series of pap ers on Graph Minors Rob ertson and Seymour proved that for

a innite set of graphs at least one member of the set is a minor of another member

This result led to the conclusion that every innite class of graphs closed under

taking minors has in its complement a nite set of forbidden substructures These

forbidden substructures or obstructions are in their smallest form graphs that are

minor minimal with resp ect to a graph prop erty P

One can construct the obstruction set of a minor closed class with a variety of

ds The following are a sample of the p opular metho ds used proven metho

If L is a minor closed set of graphs with bounded treewidth then L has

a nite set of obstructions and the following are true

L is denable by a Monadic Second Order Logic formula

L is denable by a context free graph grammar

There is a complete and safe set of reduction rules that can reduce

any graph in L to the empty graph

We have developed a set of reduction rules that can be used to help characterize

graphs with branchwidth at most four More on our results will follow in the up coming

chapter

Obstruction Sets and Treewidth

It is not uncommon to use obstruction sets to characterize graph classes

The following results have b een reached regarding the characterization of graphs

with treewidth at most k and

Rob ertson and Seymour

A graph G V E has treewidth at most if and only if G do es

not contain K as a minor

Rob ertson and Seymour

A graph G V E has treewidth at most if and only if G do es

not contain K as a minor

Arnborg and others

A graph G V E has treewidth at most if and only if G do es

not contain K M M M as a minor

Sanders expanded up on this work by computing members of the obstruction

set for graphs with treewidth at most O using k elimination sequences A k

elimination sequence for a graph G V GEG is a lab eling of its vertices

V Gv v v such that for each i i n th e degree of v is at most k

n i

in G G v v v In his PhD thesis Sanders proved that a graph

i i

has treewidth at most k if and only if it has a k elimination sequence He also con

structed over minimal forbidden minors for treewidth at most four by developing

et of reduction rules to reduce a graph with a elimination sequence down to the as

empty graph He then constructed a linear algorithm to decide if a graph G has

treewidth at most four by testing for the existence of a elimination sequence in G

Using these reductions rules for determining membership in these classes of graphs

gives way to the emb edding of a partial k tree in linear time Since currently

only the reduction rules for partial k trees where k are known it is still

considered NPhard to construct a embedding of a k tree in linear time for an arbitrary

Obstruction Sets for Bounded Branchwidth Graphs

In Graph Minors X when Rob ertson and Seymour dened the concept of branchwidth

they also gave the following prop erties of branchwidth

A graph G V E has branchwidth

ifandonly if every comp onent of G has at most one edge

at most if and only if every comp onentofG has at most one vertex

of degree at most two

at most if and only if G has no K minor

Bo dlander made an addition to the list when he determined the obstruction set of

graphs with branchwidth at most three He proved that a graph G has branc hwidth

at most

ifandonly if G has no K Q M or M minor

KQ M

53 6 M 8

Figure O

Let G V GEG be a graph and S v v v v be a subset of V G

The set S is a cross if S S v i are all minimal separators of G

i i

Bo dlaender used the fact that graphs with branchwidth at most three have graphs

that contain crosses and have treewidth at least three He used this information to

construct reduction rules to reduce a crossless graph with treewidth at most three to

an empty graph

Obstruction Sets for Other Classes of Graphs

When discussing obstruction sets it would be a ma jor oversight not to mention

Kuratowskis Theorem Kuratowskis Theorem is probably the most well known

application of Wagners Conjecture The idea that b eing able to determine if a graph

G is planar just by testing whether K or K is a minor was revolutionary at the

time and sparked a large amountofinterest in minor minimal graphs and obstruction

sets

The obstruction set for outerplanar graphs is also known Any graph with K or

K as a minor is not a outerplanar graph This leads us to another obstruction set of

jectiveplanar graph is a graph G that a class of outerplanar graphs A outerpro

canbeemb edded in the pro jective plane so that every vertex app ears on the b oundary

of a single face Archdeacon Hartseld Little and Mohar used the complete list of

the sub divisionminimal nonpro jectiveplanar graphs to show that of those

graphs are minor minimal nonouterpro jectiveplanar graphs They searched the list

of sub divisionminimal nonpro jectiveplanar graphs for the graphs which p ossessed

avertex v G such that G v is also a minorminimal nonpro jectiveplanar graph

Herein lies another metho d for determining the obstruction set One can nd the

members of a graph class H obstruction set by searching the known obstruction set

a larger graph class G where HG

Computing Obstruction Sets

on by Courcelle conjectured and later proved a result that was later extended up

others The nal result essentially stated that nite graphs with graph prop erties

that can be formulated in Monadic Second Order Logic MSOL can be decided in

p olynomial time Consequentially if there exists some unique graph prop erty that

all graphs in a graph class exhibit we can describ e it using MSOL formulas Once

that formula is available we can construct an algorithm based on our MSOL formula

that can decide in p olynomial time whether or not a graph instance is a solution to our

problem More imp ortantlysince graphs with branchwidth at most k have bounded

treewidth then we can build the obstruction set based on the MSOL formula This

result is signicant b ecause without the b ounded treewidth prop erty we can only

construct the MSOL formula once already having established the obstruction set

Bo dlaender and Thilikos proved the existence of a sentence in monadic second

order logic formula expressing whether a graph has branchwidth at most k Even

though the result and pro of are non constructive at least we know that one exists

Research done in the past to compute obstruction sets by MSOL grammar was not

computationally a go o d idea With the increase of sup ercomputing resources and

the advances in parallel computing the idea of using MSOL expressions to compute

decidable algorithms is a more viable option now

Cattell Dinneen Downey Fellows and Langston have shown that computing the

informa obstruction set of a graph prop erty P isp ossible when given the following

tion

a decision algorithm

a b ound B on the maximum treewidth of the P obstructions and

a decision algorithm for a nite index congruence that renes the

canonical congruence for P on tb oundary graphs for i B

Their metho d uses a treewidth search that essentially searches for obstructions

within the set of b ounded treewidth graphs Graphs with treewidth at most k are

constructed using unary strings and op erators Then these graphs are pruned until a

particular subgraph is found that no longer has the graph prop erty that is unique for

the treewidth b ounded class of graphs b eing considered The resulting subgraph is an

obstruction An additional condition that guarantees that the search will eventually

halt is a b ound on the number of the memb ers of the obstruction set The b ound on

the treewidth of members of the obstruction sets is used to guarantee that termination

of the algorithm will result Cattell Dinneen Downey Fellows and Langston were

successful in using this metho d to determine members of the obstruction set for graphs

with feedback vertex sets and feedback edge sets

Building up on their idea of computing obstructions we to o have implemented a

way to compute the planar members of O

Chapter

Reductions

Let G be a class of graphs A reduction R is a nite set R R where R is a

acceptable graph and R is the resulting graph after p erforming a particular sequence

of minor op erations on R Given a graph G any sequence of R complete and safe

i i

rewritings terminates with the resulting graph G being in G if and only if G is in G

A complete reduction is a reduction R on G such that for every nonempty

graph G G there is at least one reduction in R still able to be applied to G A

for a class of graphs G is a reduction such that applying any reduction safe reduction

rule to a graph G G pro duces a graph G that also is a member of G A set

of complete and safe reduction rules on a class of graphs can reduce any graph in

the class to the empty graph Unfortunately even though Arnborg et al proved

the existence of a complete and safe set of reduction rules for a class of graphs with

b ounded branchwidth the problem of constructing them for a general graph prop erty

is NPhard

Reduction Rules for Graphs with Branchwidth at Most

Three

ership in a b ounded treewidth or branchwidth class of graphs denable in Memb

Monadic Second Order Logic can be decided by nite sets of terminating reduction

rules If weknow a set of complete and safe reductions R for a class of graphs G

then any graph G for which the rep eated applications of R do not reduce G to the

empty graph shows that G is not a member of G

Bo dlaender and Thilokos were able to arrive at a complete and safe set of

reductions for graphs with branchwidth at most three They already had knowledge

of a graph prop erty containing a cross and G that identied graphs with

branchwidth at most three Using this information they develop ed a safe set of

reduction rules that reduce a graph with a cross and d own to the emptygraph

The reductions are demonstrated in Figure

Reduction Rules for Graphs with Branchwidth At Most

Four

Using this idea it was ourgoaltodevelop a complete and safe set of reduction rules

for graphs with branchwidth at most four Belowwe list a collection of safe reduction

rules that can be applied to a graph G If G is reduced to the empty graph then

G has branchwidth at most Since the set has not been proven to be complete

nothing can be said if the application of these new reduction rules coupled with the

reduction rules for graphs with branchwidth at most do not reduce a graph to the

thing we can say ab out G is that G empty set The only

Building up on Bo dlanders set of safe and complete reduction rules for graphs with

branchwidth at most three I have found three additional rules I used the idea that

any set of reduction rules for must not only reduce graphs with branchwidth at most

three down to the empty graph but also do the same for graphs with branchwidth

exactly four Bo dlander already provided the reduction rules necessary for reducing

the graphs in down to the empty graph all that is needed to do is to provide the

reduction rules that would reduce a graph with branchwidth exactly four down to

with branchwidth exactly three Afterthatis donewe can then use the reduction one

• :> 0 • :> • :> I

:> I>

Figure Reduction Rules for Graphs with G

rules R for graphs with branchwidth at most three to reduce the graphs down to

the empty graph Since every graph with branchwidth three contains K as a minor

it suces to reduce graphs with branchwidth four down to a subgraph of K

In order to accomplish this task I needed to be able to reduce every graph in

O to a subgraph of K I rst considered K the complete graph on ve vertices

The reduction rule that I discovered r works for K and also works on M another

r of O I then considered the cub e Q and found the following reduction membe

rule r that helps reduce the cub e to a subgraph of K The last reduction rule that

I needed was r which reduces M to a subgraph of K

> r1

> r2

Figure Reduction Rules r r r

These rules r r r are sequences of minor op erations Minor op erations ei

ther preserve or reduce branchwidth Consequentially these reduction rules

r rr are safe reductions when applied to graphs with branchwidth at least

four The question that needs to be determined is whether or not these reductions

reduces the branchwidth of graphs with branchwidth at least ve down to graphs

with branchwidth at most three In order to verify that these rules do not pre

vent completeness we p erformed each of the rules resp ectively in combination with

R to reduce the Petersen graph a member of O down to the empty graph

Consequently these rules in combination with all of the rules from R are a semi

complete and semisafe set of reductions In order to prove completeness a more

stronger result is needed

Once a complete and safe set of reductions is found any graph that is not reducible

to the empty graph via these reductions is a graph that has branchwidth at least ve

Thus we have away to characterize graphs with branchwidth at most four • >

0 • > •

< > I ~>I> .... ~ > +

~ > +

------> t1

Figure Reduction Rules R and r r and r

Chapter

Members of O

The b enet of the past research eorts is that they provide a guide by which one

can determine future results The work that this researcher has done in this area has

led to the discovery of several members of the obstruction set O for graphs with

branchwidth at most four

A graph is planar if and only if there exists aemb edding drawing of the graph

on a plane with no edge crossings If no crossless planar embedding of the graph can

be found then the graph is nonplanar As stated previously a graph is planar if

and only if it contains no K or K minor

following sections Wehave concentrated on computing planar memb ers of O The

outline our metho ds as well as present our results

Planar Members of O

The fact that all graphs with branchwidth at least ve have minors that b elong to

O inspired a metho d of computing a planar minor minimal subgraph with resp ect

to branchwidth at most four The results of Cattell et al verify that our metho d is

p ossible

Instead of fo cusing on the more dicult problem of nding the obstruction set

for graphs with branchwidth at most four we fo cus on an easier subproblem That

subproblem is to nd the planar memb ers of the obstruction set for graphs with

branchwidth at most four

Cattell Dinneen Downey Fellows and Langston have shown that computing the

obstruction set of a graph prop erty P isp ossible when given the following informa

tion

a decision algorithm

Using the ratcatcher algorithm and its implementation a decision algo

rithm is available to determine the branchwidth of planar graphs

a bound B on the maximum treewidth or pathwidth of the P

obstructions

This item was included so that in the pro cess of searching for members of an ob

struction set we could be more intelligent ab out which graphs we select to test If

the maximum treewidth is q than there is no need to consider graphs with treewidth

greater than q The b ounded treewidth condition is sp ecied because graphs with

treewidth at most k are easy to generate Graphs with treewidth at most k are iden

tically partial k trees Since k trees can b e easily constructed and partial k trees are

subgraphs of k trees it is easy to generate p otential graphs to test

In our particular problem our graph class has branchwidth at most four thus

trivially wehave a b ound M on the maximum treewidth of a member of our class

Instead of using the bound on treewidth to generate graphs with treewidth at most

M to run our decision algorithm on we used our conjecture that connected graphs

have branchwidth at least We used Brinkmann and McKays Plantri program to

generate planar connected graphs with varying order We only considered graphs

order b etween eight and twelve since our implementation was designed to handle with

small graphs

and a decision algorithm for a nite index congruence that renes the

canonical congruence for P on tb oundary graphs for i B

This information allows us the freedom of not having to worry ab out isomorphic

graphs b eing included in our nal set A simple isomorphism testing algorithm like

McKays Nauty program can be implemented into our algorithm to pro duce the

desired isomorphism testing comp onent of our metho d

Starting with a graph with branchwidth at least ve and p erforming various minor

op erations to reduce the resulting graph to a minor minimal subgraph with resp ect

to branchwidth four is the metho d that we used to nd planar members of O

Verifying whether the resulting graph is minor minimal consists of removing each

vertex one at a time and testing to see if the branchwidth is reduced After that

step has b een veried we p erform the same pro cedure with edge deletions and edge

contractions resp ectively

The computational dep endence of the algorithm is based up on the ratcatcher

co de that Hicks implemented The theoretical complexity is O n m With my

algorithm to verify whether or not a graph is minor minimal is O n mn m

since the ratcatcher algorithm is b eing run each time a vertex is deleted or a edge

is deleted or contracted The actual running time to verify if a graph was a minor

ect to branchwidth four lasted less than ve seconds for graphs minimal with resp

with n m To compute a minor minimal subgraph for a general graph G

could vary based on the structure of the graph as well as the size and order of the

graph For the graphs I computed since n m the running time was less

than one minute

Results

Wehave found graphs that are minor minimal planar graphs with resp ect to branch

dding so that we could have some consistency in width four We used a Fary embe

the app earance of our graphs

Figure Planar Obstruction

2 3 4

9 5

15 14

16 13

6 7

17 18

11 12

Figure Planar Obstruction

Figure Do decahedron

Figure Icosahedron

Chapter

Conclusion and Future Work

Iwas able to develop safe reduction rules for graphs with branchwidth at most four

I have b een able to compute several members of the obstruction set for graphs with

branchwidth at most four Although I have been able to compute several planar

members of O I do not know the exact b ound on howmany more planar members

need to b e computed to complete the set Although the co de itself do es not take long

to pro duce obstructions it is imp ortant to note that I have b een running the co de on

small n graphs

Analysis of the Results

A goal of this research pro ject if not to compute the entire obstruction set was to

erties that graphs must p ossess in order to have branch receive insight on the prop

width at most four Unfortunately because of the many dierent graphs that have

been computed it is dicult to pin down the general commonalities Future work

and plans will be to compute more members of O so that a more detailed and

complete analysis can o ccur

The challenge I see in continuing the work is the diculty in b eing able to generate

enough input graphs with branchwidth at least ve to result in planar memb ers that

have not already b een computed The limitations of McKays planar generation co de

is that it only allows one to sp ecify graphs with connectivity and minimum degree at

need to b e lo oked at to provide us with the input les least Thus other avenues

needed to complete the computation of the planar members of O It is imp ortant

to note that although a safe but not complete set of reductions has been revealed

in this pap er the reductions were identied b efore the members of the obstruction

set had been computed Additional safe reductions can be added to our existing set

once a more detailed analysis of the planar obstructions ensues

Potential NonPlanar Members of O

Although there is an algorithm to compute exactly the branchwidth of a planar graph

none is known for computing the branchwidth of a nonplanar graphs Thus the

metho d used to compute planar members of O can not be used to compute non

planar members of O Thus other metho ds or ideas are needed to determine what

rs of O In his dissertation Hicks showed that the Petersen the nonplanar membe

graph was a member of O He discovered this b ecause of his curiosity ab out the

branchwidth of the Peterson graph Using a similar metho d I have discovered that the

following two nonplanar are strong candidates for graphs in O These graphs that

we shall call basket and triplex are referenced by Thomas in his Excluding Minors

in Nonplanar Graphs pap er Thomas showed that every graph with minimum degree

at least three and no cycle of length less than ve has a minor isomorphic to basket

graph andor the do decahedron citethomas In other words if triplex the Petersen

F is the class of graphs with and having no cycle of length less than ve than

the obstruction set of F is basket triplex Petersen and do decahedron

Theorem Basket and Triplex have branchwidth

Pro of Basket has the cub e as a minor and triplex has M as a minor

Since b oth the cub e and M are members of O then basket and triplex

both have branchwidth at least four To even make the lower b ound

tighter we use the same tangle of the Petersen graph that Hicks used

Figure Triplex

to prove that the branchwidth of the Petersen graph was to show that

Basket and Triplex b oth have branchwidth at least ve

Separation of order

Separations of order

v G v V G

Separations of order

fv w g G v w V G

e G n e e E G

Separations of order

w u V G fv w ug G v

fv eg G n e e E Gand v V G such that v is not incident to e

P G n E P P path on nodes G

Gu v w x G n u u v w x V G such that v w x adj u

Separations of order

fv w u xg G v w u x V G

fv u eg G n e e E Gand v u V G such that v and u are not

incident to e

v P G n E P P path on no des G and v V G such that v is

notapart of P

P G n E P P path on nodes G

fe f g G nfe f g pairs e f E G such that e and f do not share a no de

Gu v w x z G n E Gu v w x z u u v w x z V G such that

v w x adj u

Guv adj uadj v G nfu v g uv E G

Figure Tangle of order for b oth Basket and Triplex

No w that is all left to show is that we can pro duce a branch decomp osition

of b oth Basket and Triplex that has width ve The following is provided

below

In order to show that these are minor minimal with resp ect to branc hwidth at

most four we need to show that every minor of basket and triplex has branchwidth

1 9 10

17 2 8

11 13 12 15

7 3

14 16

6 4

5 3 4 18

1 2 11 8 15

16 13

5 12 17 14 7 6

Figure Triplex and a branch decomp osition of width

8 1

9 11 10 2 17 7 16 12 13 3 15 14 6

5 4

14 10 12 16

18 7 5 15 3

9 17 6 4

Figure Basket and a branch decomp osition of width

at most four Unlike the Petersen graph which has all minors isomorphic to one

of two certain graphs there is not a similar result for the basket and triplex graph

In order to verify that basket and triplex is minor minimal we need to show that

for any vertex deletion edge deletion and edge contraction the resulting graph has

branchwidth at most four This can be shown by providing a branch decomp osition

of width at most four Taking advantage of the symmetry of basket and triplex there

are only three types of vertices and three types of edges which need to b e considered

As of yet I have b een able to proven that for the basket graph deletion of anyvertex

pro duces a graph which has branchwidth at most four and deletion of two types of

edges pro duces a graph with branchwidth at most four What is left to show is to

complete the result for all edge deletions and all edge contractions

Bibliography

H L Bo dlaender and D M Thilkos Constructive linear time algorithms for

branchwidth Lecture Notes in Computer Science

HL Bo dlaender On linear time minor tests with depthrst search Journal of

Algorithms pages

HL Bo dlaender Tutorial A partial k arb oretum of graphs with bounded

treewidth Theoretical Computer Science

K Cattell M Dinneen R DowneyMFellows and M Langston On computing

graph minor obstruction sets The Journal of Universal Computer Science

K Cattell M Dinneen and M Fellows Obstructions to within a few vertices

or edges of acyclic

sets of B Courcelle The monadic secondorder logic of graphs i Recognizable

nite graphs Information and Computation

B Courcelle and G Senizergues The obstructions of a minorclosed set of graphs

dened by a context free grammar Discrete Mathematics

CHC Little D Archdeacon N Hartseld and B Mohar Obstruction sets for

outerpro jectiveplanar graphs Ars Combinatoria

I Hicks Branch decompositions and their applications PhD thesis Rice Uni

versity

Andrzej Proskurowski Graph reductions and techniques for nding minimal

forbidden minors Contemporary Mathematics

N Rob ertson and PD Seymour Graph minors a survey Surveys in combina

torics

N Rob ertson and PD Seymour Graph minors algorithmic asp ects of

treewidth Journal of Algorithms

N Rob ertson and PD Seymour Graph minors obstructions to tree decom

p ositions Journal of Combinatorial Theory Series B

A Proskurowski S Arnb org B Courcelle and D Seese An algebraic theory of

graph reduction Journal of ACM

D Corneil S Arnborg A Proskurowski Forbidden minors characterization of

partial trees Discrete Mathematics

D Sanders Linear Algorithms for Graphs of Treewidth at Most Four PhD

thesis Georgia Tech

D Sanders On linear recognition of tree width at most four Journal of Algo

rithms

PD Seymour and R Thomas Call routing and the ratcatcher Combinatorica