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On Treewidth and Graph Minors

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On Treewidth and Graph Minors

Daniel John Harvey

Submitted in total fulfilment of the requirements of the degree of
Doctor of Philosophy
February 2014

Department of Mathematics and Statistics
The University of Melbourne

Produced on archival quality paper

ii

Abstract

Both treewidth and the Hadwiger number are key graph parameters in structural and algorithmic graph theory, especially in the theory of graph minors. For example, treewidth demarcates the two major cases of the Robertson and Seymour proof of Wagner’s Conjecture. Also, the Hadwiger number is the key measure of the structural complexity of a graph. In this thesis, we shall investigate these parameters on some interesting classes of graphs.
The treewidth of a graph defines, in some sense, how “tree-like” the graph is. Treewidth is a key parameter in the algorithmic field of fixed-parameter tractability. In particular, on classes of bounded treewidth, certain NP-Hard problems can be solved in polynomial time. In structural graph theory, treewidth is of key interest due to its part in the stronger form of Robertson and Seymour’s Graph Minor Structure Theorem. A key fact is that the treewidth of a graph is tied to the size of its largest grid minor. In fact, treewidth is tied to a large number of other graph structural parameters, which this thesis thoroughly investigates. In doing so, some of the tying functions between these results are improved. This thesis also determines exactly the treewidth of the line graph of a complete graph. This is a critical example in a recent paper of Marx, and improves on a recent result by Grohe and Marx. By extending the techniques used, we also determine the treewidth of the line graph of a complete multipartite graph, up to lower order terms in general, and exactly whenever the complete multipartite graph is regular. This generalises a result by Lucena. We also determine a lower bound on the treewidth of any line graph; this result is similar to a question about the Hadwiger number of line graphs posed by Seymour, which was recently proven by DeVos et al.. Finally, we prove a result on the treewidth of the Kneser graph; in doing so we also prove a generalisation of the famous Erdo˝s-Ko-Rado Theorem.
The Hadwiger number of a graph is the size of its largest complete minor. One of the most important conjectures in modern mathematics is Hadwiger’s Conjecture, which conjectures that the Hadwiger number of a graph is at least its chromatic number. A related question is determining what lower bound on the average degree is required to

iii iv

ABSTRACT

ensure the existence of a Kt-minor (or, more generally, an H-minor for any graph H). The Kt-minor case has been thoroughly studied, and independently answered by Kostochka and Thomason. In this thesis we answer a slightly different question and present an algorithm for finding, in O(n) time, an H-minor forced by high average degree. Finally, this thesis determines a weakening of Hadwiger’s Conjecture on the class of circular arc graphs, an interesting generalisation of the class of interval graphs, and in the process of doing so, proves some useful results about linkages in interval graphs.

Declaration

This is to certify that:
(i) the thesis comprises only my original work towards the PhD except where indicated in the Preface,

(ii) due acknowledgement has been made in the text to all other material used, and
(iii) the thesis is less that 100,000 words in length, exclusive of tables, maps, bibliographies and appendices.

vvi

DECLARATION

Preface

No work presented in this thesis has been submitted for any other kind of qualification, and no work in this thesis was carried out prior to PhD enrolment.
Other than in Chapter 1 and Chapter 2, all results presented are new. (Some of the results presented in Chapter 2, an investigation into parameters tied to treewidth, are improvements on previously presented results. These results are noted in the text.)
Chapters 2,3,4,5,6 and 8 were carried out in collaboration with my supervisor, David
Wood. Chapter 7 is the result of joint work with Vida Dujmovi´c, Gwena¨el Joret, Bruce Reed and David Wood. An important example in Chapter 5 was provided by Bruce Reed.
The results of Chapter 7 extend results which have been recently published in the SIAM
Journal on Discrete Mathematics [26]. The results in Chapter 6 have recently appeared in the Electronic Journal of Combinatorics [44]. The results in Chapter 3 have been accepted by the Journal of Graph Theory [45]. (An earlier version of this paper, including the results of Chapter 4, is available on the arXiv [42].) Chapter 2 is also available on the arXiv [43].
In all cases I, Daniel Harvey, was the primary and corresponding author.

vii viii

PREFACE

Acknowledgements

I’d like to acknowledge the following individuals for their assistance during my PhD candidature. Firstly, I’d like to thank my supervisor David Wood for the enormous amount of assistance provided throughout my time as a PhD student.
For their guidance, I’d like to thank the members of my Advisory Committee—Sanming
Zhou, Peter Forrester and Graham Farr.
I’d like to thank my collaborators Vida Dujmovi´c, Gwena¨el Joret and Bruce Reed for their work on the paper which would become Chapter 7.
Further thanks to my officemates and fellow students Michael Payne, Guangjun Xu and Ricky Rotheram for their advice and support.
I also wish to acknowledge the several reading and seminar groups that ran during my candidature. Firstly, the Graph Theory Reading Group, and its current leader Arun Mani. Secondly, the Discrete Structures and Algorithms seminar group. Finally, the Theoretical Research in Computer Science (TRICS) group, run by Tony Wirth and (previously) Kerri Morgan.
Thanks to Alex Scott for pointing out references [36, 37, 99] in Chapter 6. Thanks also to Jacob Fox for helpful conversations with regards to Chapter 2.
A final note of thanks to my parents David and Jan Harvey for their emotional support during the last four years.

ix x

ACKNOWLEDGEMENTS

Table of Contents

  • Abstract
  • iii

  • v
  • Declaration

  • Preface
  • vii

  • ix
  • Acknowledgements

Table of Contents List of Figures xi xv

  • 1 Introduction and Literature Review
  • 1

13
1.1 Graph Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Treewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Hadwiger’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 A Unifying Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

  • I
  • Treewidth
  • 19

  • 2 Parameters Tied to Treewidth
  • 21

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Brambles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 k-Trees and Chordal Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 Separators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.6 Branchwidth and Tangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.7 Tree Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.8 Linkedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.9 Well-linked and k-Connected Sets . . . . . . . . . . . . . . . . . . . . . . . . 38

xi xii

TABLE OF CONTENTS

2.10 Grid Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.11 Grid-like Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.12 Fractional Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

  • 3 Treewidth of the Line Graph of a Complete Graph
  • 45

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Line-Brambles and the Treewidth Duality Theorem . . . . . . . . . . . . . . 46 3.3 Proof of Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

  • 4 Treewidth of the Line Graph of a Complete Multipartite Graph
  • 51

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 Line-Brambles of a Complete Multipartite Graph . . . . . . . . . . . . . . . 52 4.3 Path Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

  • 5 Treewidth of General Line Graphs
  • 73

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2 The General Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.3 The General Upper Bound and Extensions . . . . . . . . . . . . . . . . . . . 79

  • 6 Treewidth of the Kneser Graph and the Erd˝os-Ko-Rado Theorem
  • 81

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.2 Basic Definitions and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 82 6.3 Upper Bound for Treewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.4 Separators in the Kneser Graph . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.5 Lower Bound for Treewidth when k = 2 . . . . . . . . . . . . . . . . . . . . 91 6.6 A Weaker Lower Bound for Treewidth . . . . . . . . . . . . . . . . . . . . . 92 6.7 Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

  • II Graph Minors
  • 99

  • 7 Finding a Minor Quickly in Graphs with High Average Degree
  • 101

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.3 Correctness of Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.4 Time Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

  • 8 Hadwiger’s Conjecture for Circular Arc Graphs
  • 105

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

TABLE OF CONTENTS

xiii
8.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 8.3 Special Path Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 8.4 Colouring G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 8.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

  • 9 Linkages in Interval Graphs
  • 123

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 9.2 “Selection Sort” Paths in the Power of a Path . . . . . . . . . . . . . . . . . 125 9.3 Improved Linkages in Interval Graphs . . . . . . . . . . . . . . . . . . . . . 128 9.4 Hadwiger Number of the Power of a Cycle . . . . . . . . . . . . . . . . . . . 135

Bibliography Index
137 146

xiv

TABLE OF CONTENTS

List of Figures

  • 1.1 An example graph and tree decomposition. . . . . . . . . . . . . . . . . . .
  • 4

1.2 Catlin’s counterexample to Haj´os’ Conjecture. . . . . . . . . . . . . . . . . . 14 1.3 C92 represented as a circular arc graph. . . . . . . . . . . . . . . . . . . . . . 17

2.1 The graph ψ4,2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.1 The described path decomposition for L(K6). . . . . . . . . . . . . . . . . . 50 4.1 A red ordering for L(K5,2). . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 A path decomposition for L(K5,2) using the red ordering. . . . . . . . . . . 65 4.3 A blue ordering for L(K2,2,2). . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.4 A path decomposition for L(K2,2,2) using the blue ordering. . . . . . . . . . 68

5.1 An example graph and a tree decomposition of its line graph. . . . . . . . . 76 5.2 A diagram of Lemma 5.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.1 The graph Kneser(5, 2) and a tree decomposition of Kneser(5, 2). . . . . . . 85 6.2 A tree decomposition of Kneser(n, k). . . . . . . . . . . . . . . . . . . . . . . 86 6.3 Diagram for Claim 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.4 Diagram for Claim 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.5 Diagram for Claim 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.6 Diagram for Claim 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

8.1 An illustration of Case 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8.2 An illustration of Case 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

9.1 Connecting the sources to the vertices left of si. . . . . . . . . . . . . . . . . 125 9.2 Diagram for Lemma 9.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 9.3 The first case for Lemma 9.9 . . . . . . . . . . . . . . . . . . . . . . . . . . 132 9.4 The second case for Lemma 9.9 . . . . . . . . . . . . . . . . . . . . . . . . . 133

xv xvi

LIST OF FIGURES

Chapter 1

Introduction and Literature Review

1.1 Graph Minors

The numbered Theorems 1.1 to 1.13 form the core results presented in this thesis. In this chapter we present the statements of these theorems and provide the necessary background and context; the remaining chapters contain thorough proofs of these results.
A graph G is a set of elements V (G), called vertices, together with a set E(G) of unordered pairs of vertices, called edges. This is the definition of a simple graph; for our purposes, all graphs are simple unless otherwise noted. All of the basic definitions and notation in this thesis are as in the standard textbook by Diestel [22], unless otherwise noted.
A graph H is a minor of a graph G if a graph isomorphic to H can be constructed

from G by repeated applications of vertex deletion, edge deletion and edge contraction.

Vertex deletion and edge deletion are self-evident. Edge contraction is defined as follows: given an edge vw ∈ E(G), replace vertices v and w with a single new vertex x adjacent to all vertices initially adjacent to v or w. If H is a minor of G, we say that G contains an H-minor. The study of graph minors is of key importance in modern graph theory. We now discuss some of these key results.
An important early result is the Kuratowski-Wagner Theorem [66, 113]. A graph is planar if it can be drawn in the plane so that no two edges cross.

Theorem (Kuratowski-Wagner Theorem [113]). A graph G is planar if and only if it contains neither K5 nor K3,3 as a minor.

We say a class of graphs G is minor-closed if H ∈ G whenever G ∈ G and H is a
1
2

CHAPTER 1. INTRODUCTION

minor of G. The class of planar graphs is minor-closed; clearly vertex and edge deletion do not turn a planar graph non-planar, and edge contraction does not create any crossing edges since it is possible to move two vertices together by “shortening” the edge until they merge together. The Kuratowski-Wagner Theorem gives a finite forbidden minor characterisation of the planar graphs—that is, a finite set of graphs such that every graph that is non-planar must contain a minor in that set, and every graph that is planar contains no minor in that set. While it is easy to find an infinite set of graphs with this property (simply take all graphs not in G), it is not obvious that a finite set must exist.
A similar result holds for every non-trivial minor-closed class.

Theorem (Graph Minor Theorem [88]). Let G be a minor-closed class of graphs, other than the class of all graphs. Then G has a finite forbidden minor characterisation.

This is one of the key theorems in modern structural graph theory. It was originally referred to as Wagner’s Conjecture, before being proven by Robertson and Seymour in their twenty-three paper sequence “Graph Minors” [88]. (Wagner, however, stated that he had never conjectured such a result [22].) The Graph Minor Theorem gives rise to a O(n3) time algorithm for determining whether or not a given n-vertex graph G is in a fixed minor-closed class G [93]. Specifically, this algorithm determines whether G contains a fixed graph H as a minor in O(n3) time; given a forbidden minor characterisation for G, it is sufficient to check whether or not each graph is a minor of G. However, this algorithm requires that the finite forbidden minor characterisation be known for G, which is not the case for most classes. It also depends on the number of graphs in this characterisation. This is finite and does not depend on n (which is why the total algorithm is still only O(n3) time; there is one iteration of the loop for each graph in the characterisation), but it is often enormous (see [13], for example).
At the heart of Robertson and Seymour’s proof is the Graph Minor Structure Theorem
[91, 94]. There are several different versions of the Graph Minor Structure Theorem (see Kawarabayashi and Mohar [53] for an overview), but essentially it shows that a minorclosed class either has bounded treewidth (which we define in Section 1.2), or if the class has unbounded treewidth, then any graph in it can be constructed with a restricted series of operations. A simplified version of this theorem follows:

Theorem (Robertson and Seymour [94]). Let H be a non-planar graph and let G be the family of graphs with no H-minor, and denote by Σ1, . . . , Σs all the connected surfaces (up to homeomorphism) in which H cannot be drawn. Then every graph in G can be constructed by clique-sums from those that can “almost” be drawn in some Σi.

Note the following facts. Firstly, we shall deal with the “missing” case when H is

1.2. TREEWIDTH

3planar in Section 1.2. Secondly, by “almost”, we mean that if G can “almost” be drawn in Σi, then G is embeddable in Σi except for a small number of apex vertices and a small number of vortices. Apex vertices are allowed to be adjacent to any other vertex without their incident edges “counting” with respect to the embedding. Vortices are subgraphs of G with bounded pathwidth (again defined in Section 1.2), which meet Σi without intersecting each other or other vertices except in a very restricted way. By “a small number”, we mean depending only on H, and not on G or |G| itself. Thirdly, a graph G is a clique sum of graphs G1 and G2 if G can be constructed by choosing cliques of equal size in G1 and G2, identifying them, and then possibly deleting some edges from the identified clique. Finally, note that this version of the Graph Minor Structure Theorem was insufficient for Robertson and Seymour to use to obtain their result, as they explain in [94]. The stronger version of the Graph Minor Structure Theorem uses the concept of tangles. We discuss tangles in Section 2.6.
A key parameter in the Graph Minor Structure Theorem, and the piece of the puzzle most interesting to us, is the parameter known as treewidth.

1.2 Treewidth

Let G be a graph. A tree decomposition of G is a pair (T, (Bx ⊆ V (G))x∈V (T)) consisting of

• a tree T, and • a collection of bags Bx containing vertices of G, indexed by the nodes of T,

such that:
• for all v ∈ V (G), the set {x ∈ V (T) : v ∈ Bx} induces a non-empty subtree of T, and
• for all vw ∈ E(G), there is some bag Bx containing both v and w.

The width of a tree decomposition is the size of the largest bag in the tree decomposition, minus 1. The treewidth of a graph, denoted by tw(G), is the minimum width over all tree decompositions of G. To illustrate these concepts, we provide an example tree decomposition in Figure 1.1. Often, for the sake of simplicity, we will refer to a tree decomposition simply as T, leaving the set of bags implied whenever this is unambiguous. For similar reasons, often we say that bags X and Y are adjacent (or we refer to an edge XY ), instead of the more accurate statement that the nodes of T indexing X and Y are adjacent. Usually, we refer to the vertices of G as vertices, but refer to the vertices of T as nodes, as in the above definition. This is also done to avoid confusion. 4

CHAPTER 1. INTRODUCTION

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  • Lecture 10: April 20, 2005 Perfect Graphs

    Lecture 10: April 20, 2005 Perfect Graphs

    Re-revised notes 4-22-2005 10pm CMSC 27400-1/37200-1 Combinatorics and Probability Spring 2005 Lecture 10: April 20, 2005 Instructor: L´aszl´oBabai Scribe: Raghav Kulkarni TA SCHEDULE: TA sessions are held in Ryerson-255, Monday, Tuesday and Thursday 5:30{6:30pm. INSTRUCTOR'S EMAIL: [email protected] TA's EMAIL: [email protected], [email protected] IMPORTANT: Take-home test Friday, April 29, due Monday, May 2, before class. Perfect Graphs k 1=k Shannon capacity of a graph G is: Θ(G) := limk (α(G )) : !1 Exercise 10.1 Show that α(G) χ(G): (G is the complement of G:) ≤ Exercise 10.2 Show that χ(G H) χ(G)χ(H): · ≤ Exercise 10.3 Show that Θ(G) χ(G): ≤ So, α(G) Θ(G) χ(G): ≤ ≤ Definition: G is perfect if for all induced sugraphs H of G, α(H) = χ(H); i. e., the chromatic number is equal to the clique number. Theorem 10.4 (Lov´asz) G is perfect iff G is perfect. (This was open under the name \weak perfect graph conjecture.") Corollary 10.5 If G is perfect then Θ(G) = α(G) = χ(G): Exercise 10.6 (a) Kn is perfect. (b) All bipartite graphs are perfect. Exercise 10.7 Prove: If G is bipartite then G is perfect. Do not use Lov´asz'sTheorem (Theorem 10.4). 1 Lecture 10: April 20, 2005 2 The smallest imperfect (not perfect) graph is C5 : α(C5) = 2; χ(C5) = 3: For k 2, C2k+1 imperfect.
  • Arxiv:2106.16130V1 [Math.CO] 30 Jun 2021 in the Special Case of Cyclohedra, and by Cardinal, Langerman and P´Erez-Lantero [5] in the Special Case of Tree Associahedra

    Arxiv:2106.16130V1 [Math.CO] 30 Jun 2021 in the Special Case of Cyclohedra, and by Cardinal, Langerman and P´Erez-Lantero [5] in the Special Case of Tree Associahedra

    LAGOS 2021 Bounds on the Diameter of Graph Associahedra Jean Cardinal1;4 Universit´elibre de Bruxelles (ULB) Lionel Pournin2;4 Mario Valencia-Pabon3;4 LIPN, Universit´eSorbonne Paris Nord Abstract Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph G encodes the combinatorics of search trees on G, defined recursively by a root r together with search trees on each of the connected components of G − r. In particular, the skeleton of the graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. It is known that the diameter of the associahedra of paths of length n, the classical associahedra, is 2n − 6 for a large enough n. We give a tight bound of Θ(m) on the diameter of trivially perfect graph associahedra on m edges. We consider the maximum diameter of associahedra of graphs on n vertices and of given tree-depth, treewidth, or pathwidth, and give lower and upper bounds as a function of these parameters. Finally, we prove that the maximum diameter of associahedra of graphs of pathwidth two is Θ(n log n). Keywords: generalized permutohedra, graph associahedra, tree-depth, treewidth, pathwidth 1 Introduction The vertices and edges of a polyhedron form a graph whose diameter (often referred to as the diameter of the polyhedron for short) is related to a number of computational problems. For instance, the question of how large the diameter of a polyhedron arises naturally from the study of linear programming and the simplex algorithm (see, for instance [27] and references therein).
  • Average Degree Conditions Forcing a Minor∗

    Average Degree Conditions Forcing a Minor∗

    Average degree conditions forcing a minor∗ Daniel J. Harvey David R. Wood School of Mathematical Sciences Monash University Melbourne, Australia [email protected], [email protected] Submitted: Jun 11, 2015; Accepted: Feb 22, 2016; Published: Mar 4, 2016 Mathematics Subject Classifications: 05C83 Abstract Mader first proved that high average degree forces a given graph as a minor. Often motivated by Hadwiger's Conjecture, much research has focused on the av- erage degree required to force a complete graph as a minor. Subsequently, various authors have considered the average degree required to force an arbitrary graph H as a minor. Here, we strengthen (under certain conditions) a recent result by Reed and Wood, giving better bounds on the average degree required to force an H-minor when H is a sparse graph with many high degree vertices. This solves an open problem of Reed and Wood, and also generalises (to within a constant factor) known results when H is an unbalanced complete bipartite graph. 1 Introduction Mader [13, 14] first proved that high average degree forces a given graph as a minor1. In particular, Mader [13, 14] proved that the following function is well-defined, where d(G) denotes the average degree of a graph G: f(H) := inffD 2 R : every graph G with d(G) > D contains an H-minorg: Often motivated by Hadwiger's Conjecture (see [18]), much research has focused on f(Kt) where Kt is the complete graph on t vertices. For 3 6 t 6 9, exact bounds on the number of edges due to Mader [14], Jørgensen [6], and Song and Thomas [19] implyp that f(Kt) = 2t−4.
  • Empirical Comparison of Greedy Strategies for Learning Markov Networks of Treewidth K

    Empirical Comparison of Greedy Strategies for Learning Markov Networks of Treewidth K

    Empirical Comparison of Greedy Strategies for Learning Markov Networks of Treewidth k K. Nunez, J. Chen and P. Chen G. Ding and R. F. Lax B. Marx Dept. of Computer Science Dept. of Mathematics Dept. of Experimental Statistics LSU LSU LSU Baton Rouge, LA 70803 Baton Rouge, LA 70803 Baton Rouge, LA 70803 Abstract—We recently proposed the Edgewise Greedy Algo- simply one less than the maximum of the orders of the rithm (EGA) for learning a decomposable Markov network of maximal cliques [5], a treewidth k DMN approximation to treewidth k approximating a given joint probability distribution a joint probability distribution of n random variables offers of n discrete random variables. The main ingredient of our algorithm is the stepwise forward selection algorithm (FSA) due computational benefits as the approximation uses products of to Deshpande, Garofalakis, and Jordan. EGA is an efficient alter- marginal distributions each involving at most k + 1 of the native to the algorithm (HGA) by Malvestuto, which constructs random variables. a model of treewidth k by selecting hyperedges of order k +1. In The treewidth k DNM learning problem deals with a given this paper, we present results of empirical studies that compare empirical distribution P ∗ (which could be given in the form HGA, EGA and FSA-K which is a straightforward application of FSA, in terms of approximation accuracy (measured by of a distribution table or in the form of a set of training data), KL-divergence) and computational time. Our experiments show and the objective is to find a DNM (i.e., a chordal graph) that (1) on the average, all three algorithms produce similar G of treewidth k which defines a probability distribution Pµ approximation accuracy; (2) EGA produces comparable or better that ”best” approximates P ∗ according to some criterion.
  • Minimal Acyclic Forbidden Minors for the Family of Graphs with Bounded Path-Width

    Minimal Acyclic Forbidden Minors for the Family of Graphs with Bounded Path-Width

    Discrete Mathematics 127 (1994) 293-304 293 North-Holland Minimal acyclic forbidden minors for the family of graphs with bounded path-width Atsushi Takahashi, Shuichi Ueno and Yoji Kajitani Department of Electrical and Electronic Engineering, Tokyo Institute of Technology. Tokyo, 152. Japan Received 27 November 1990 Revised 12 February 1992 Abstract The graphs with bounded path-width, introduced by Robertson and Seymour, and the graphs with bounded proper-path-width, introduced in this paper, are investigated. These families of graphs are minor-closed. We characterize the minimal acyclic forbidden minors for these families of graphs. We also give estimates for the numbers of minimal forbidden minors and for the numbers of vertices of the largest minimal forbidden minors for these families of graphs. 1. Introduction Graphs we consider are finite and undirected, but may have loops and multiple edges unless otherwise specified. A graph H is a minor of a graph G if H is isomorphic to a graph obtained from a subgraph of G by contracting edges. A family 9 of graphs is said to be minor-closed if the following condition holds: If GEB and H is a minor of G then H EF. A graph G is a minimal forbidden minor for a minor-closed family F of graphs if G#8 and any proper minor of G is in 9. Robertson and Seymour proved the following deep theorems. Theorem 1.1 (Robertson and Seymour [15]). Every minor-closed family of graphs has a finite number of minimal forbidden minors. Theorem 1.2 (Robertson and Seymour [14]).
  • Arxiv:2006.06067V2 [Math.CO] 4 Jul 2021

    Arxiv:2006.06067V2 [Math.CO] 4 Jul 2021

    Treewidth versus clique number. I. Graph classes with a forbidden structure∗† Cl´ement Dallard1 Martin Milaniˇc1 Kenny Storgelˇ 2 1 FAMNIT and IAM, University of Primorska, Koper, Slovenia 2 Faculty of Information Studies, Novo mesto, Slovenia [email protected] [email protected] [email protected] Treewidth is an important graph invariant, relevant for both structural and algo- rithmic reasons. A necessary condition for a graph class to have bounded treewidth is the absence of large cliques. We study graph classes closed under taking induced subgraphs in which this condition is also sufficient, which we call (tw,ω)-bounded. Such graph classes are known to have useful algorithmic applications related to variants of the clique and k-coloring problems. We consider six well-known graph containment relations: the minor, topological minor, subgraph, induced minor, in- duced topological minor, and induced subgraph relations. For each of them, we give a complete characterization of the graphs H for which the class of graphs excluding H is (tw,ω)-bounded. Our results yield an infinite family of χ-bounded induced-minor-closed graph classes and imply that the class of 1-perfectly orientable graphs is (tw,ω)-bounded, leading to linear-time algorithms for k-coloring 1-perfectly orientable graphs for every fixed k. This answers a question of Breˇsar, Hartinger, Kos, and Milaniˇc from 2018, and one of Beisegel, Chudnovsky, Gurvich, Milaniˇc, and Servatius from 2019, respectively. We also reveal some further algorithmic implications of (tw,ω)- boundedness related to list k-coloring and clique problems. In addition, we propose a question about the complexity of the Maximum Weight Independent Set prob- lem in (tw,ω)-bounded graph classes and prove that the problem is polynomial-time solvable in every class of graphs excluding a fixed star as an induced minor.
  • Arxiv:1602.02002V1 [Math.CO] 5 Feb 2016 Dept

    Arxiv:1602.02002V1 [Math.CO] 5 Feb 2016 Dept

    a The Structure of W4-Immersion-Free Graphs R´emy Belmonteb;c Archontia Giannopouloud;e Daniel Lokshtanovf ;g Dimitrios M. Thilikosh;i ;j Abstract We study the structure of graphs that do not contain the wheel on 5 vertices W4 as an immersion, and show that these graphs can be constructed via 1, 2, and 3-edge-sums from subcubic graphs and graphs of bounded treewidth. Keywords: Immersion Relation, Wheel, Treewidth, Edge-sums, Structural Theorems 1 Introduction A recurrent theme in structural graph theory is the study of specific properties that arise in graphs when excluding a fixed pattern. The notion of appearing as a pattern gives rise to various graph containment relations. Maybe the most famous example is the minor relation that has been widely studied, in particular since the fundamental results of Kuratowski and Wagner who proved that planar graphs are exactly those graphs that contain neither K5 nor K3;3 as a (topological) minor. A graph G contains a graph H as a topological minor if H can be obtained from G by a sequence of vertex deletions, edge deletions and replacing internally vertex-disjoint paths by single edges. Wagner also described the structure of the graphs that exclude K5 as a minor: he proved that K5- minor-free graphs can be constructed by \gluing" together (using so-called clique-sums) planar graphs and a specific graph on 8 vertices, called Wagner's graph. aEmails of authors: [email protected] [email protected], [email protected], [email protected] b arXiv:1602.02002v1 [math.CO] 5 Feb 2016 Dept.
  • Handout on Vertex Separators and Low Tree-Width K-Partition

    Handout on Vertex Separators and Low Tree-Width K-Partition

    Handout on vertex separators and low tree-width k-partition January 12 and 19, 2012 Given a graph G(V; E) and a set of vertices S ⊂ V , an S-flap is the set of vertices in a connected component of the graph induced on V n S. A set S is a vertex separator if no S-flap has more than n=p 2 vertices. Lipton and Tarjan showed that every planar graph has a separator of size O( n). This was generalized by Alon, Seymour and Thomas to any family of graphs that excludes some fixed (arbitrary) subgraph H as a minor. Theorem 1 There a polynomial time algorithm that given a parameter h and an n vertex graphp G(V; E) either outputs a Kh minor, or outputs a vertex separator of size at most h hn. Corollary 2 Let G(V; E) be an arbitrary graph with nop Kh minor, and let W ⊂ V . Then one can find in polynomial time a set S of at most h hn vertices such that every S-flap contains at most jW j=2 vertices from W . Proof: The proof given in class for Theorem 1 easily extends to this setting. 2 p Corollary 3 Every graph with no Kh as a minor has treewidth O(h hn). Moreover, a tree decomposition with this treewidth can be found in polynomial time. Proof: We have seen an algorithm that given a graph of treewidth p constructs a tree decomposition of treewidth 8p. Usingp Corollary 2, that algorithm can be modified to give a tree decomposition of treewidth 8h hn in our case, and do so in polynomial time.
  • The Structure of Graphs Not Topologically Containing the Wagner Graph

    The Structure of Graphs Not Topologically Containing the Wagner Graph

    The structure of graphs not topologically containing the Wagner graph John Maharry, Neil Robertson1 Department of Mathematics The Ohio State University Columbus, OH USA [email protected], [email protected] Abstract A structural characterization of graphs not containing the Wagner graph, also known as V8, is shown. The result was announced in 1979 by the second author, but until now a proof has not been published. Keywords: Wagner graph, V8 graph, Excluded-minor 1. Introduction Graphs in this paper are finite and have no loops or multiple edges. Given a graph G, one can obtain a contraction K of G by contracting pairwise dis- joint connected induced subgraphs to single (distinct) vertices where distinct vertices of K are adjacent if and only if there exists an edge of G with an end- vertex in each of the corresponding subgraphs of G. A graph M is a minor of a graph G when M is a contraction of a subgraph of G. We write H ≤m G when H is isomorphic to a minor M of G. Given an edge e 2 E(G) with endvertices x and y, we define the single-edge contraction to be the graph formed by contracting the edge e to a new vertex, call it z. An equivalent definition of minor inclusion of a graph is H ≤m G if and only if H is iso- morphic to a graph obtained by a series of single-edge contractions starting with a subgraph S of G. 1Research funded in part by King Abdulaziz University, 2011 and in part by SERC Visiting Fellowship Research Grant, University of London, 1985 Preprint submitted to Journal Combin.