On excluded minors and biased graph representations of frame matroids by Daryl Funk M.Sc., University of Victoria, 2009 B.Sc., Simon Fraser University, 1992 Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Department of Mathematics Faculty of Science c Daryl Funk 2015 SIMON FRASER UNIVERSITY Spring 2015 All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced without authorization under the conditions for “Fair Dealing.” Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review and news reporting is likely to be in accordance with the law, particularly if cited appropriately. APPROVAL Name: Daryl Funk Degree: Doctor of Philosophy (Mathematics) Title of Thesis: On excluded minors and biased graph representations of frame matroids Examining Committee: Dr. Jonathan Jedwab, Chair Professor, Department of Mathematics Dr. Matthew DeVos Senior Supervisor Associate Professor, Department of Mathematics Dr. Luis Goddyn Co-Supervisor Professor, Department of Mathematics Dr. Bojan Mohar Internal Examiner Professor, Department of Mathematics Dr. Daniel Slilaty External Examiner Professor, Department of Mathematics and Statistics Wright State University Date Defended: 8 January 2015 ii Partial Copyright Licence iii ABSTRACT A biased graph is a graph in which every cycle has been given a bias, either balanced or unbalanced. Biased graphs provide representations for an important class of matroids, the frame matroids. As with graphs, we may take minors of biased graphs and of matroids, and a family of biased graphs or matroids is minor-closed if it contains every minor of every member of the family. For any such class, we may ask for the set of those objects that are minimal with respect to minors subject to not belonging to the class — i.e., we may ask for the set of excluded minors for the class. A frame matroid need not be uniquely represented by a biased graph. This creates complications for the study of excluded minors. Hence this thesis has two main intertwining lines of investigation: (1) excluded minors for classes of frame matroids, and (2) biased graph representations of frame matroids. Trying to determine the biased graphs representing a given frame matroid leads to the necessity of determining the biased graphs representing a given graphic matroid. We do this in Chapter 3. Determining all possible biased graph representations of non-graphic frame matroids is more difficult. In Chapter 5 we determine all biased graphs representa- tions of frame matroids having a biased graph representation of a certain form, subject to an additional connectivity condition. Perhaps the canonical examples of biased graphs are group-labelled graphs. Not all biased graphs are group-labellable. In Chapter 2 we give two characterisations of those biased graphs that are group labellable, one topological in nature and the other in terms of the existence of a sequence of closed walks in the graph. In contrast to graphs, which are well-quasi-ordered by the minor relation, this characterisation enables us to construct infinite antichains of biased graphs, even with each member on a fixed number of vertices. These constructions are then used to exhibit infinite antichains of frame matroids, each of whose members are of a fixed rank. In Chapter 4, we begin an investigation of excluded minors for the class of frame ma- troids by seeking to determine those excluded minors that are not 3-connected. We come close, determining a set E of 18 particular excluded minors and drastically narrowing the search for any remaining such excluded minors. Keywords: Frame matroid; biased graph; excluded minors; representations; group-labelling; gain graph; well-quasi-ordering; lift matroid; graphic matroid iv I am asked sometimes what a matroid is. I often “You put it very graphically," said Humpty revert to our sacred writings and recall the en- Dumpty. counter of Alice with the grinning Cheshire cat. At one stage the cat vanishes away, beginning “But I can’t do it with all the grins," said Alice. with the tip of its tail and ending with the grin, “Some of them have the most uncatly shapes. which persists long after the remainder of the Whatever can be behind them?" cat. “That’s what makes it interesting," said Humpty “I expect you saw a lot of loose grins wandering Dumpty. “You have to classify the Uncats. around," said Humpty Dumpty. “Yes, indeed," said Alice. “But with some of them you could see they belonged to cats. I It now will be right to describe kept trying to imagine what the cats behind them Each particular batch were like." Distinguishing those that are Fanos and bite “What an Auslandish thing to do," said Humpty From K. Kuratowskis that scratch." Dumpty. “Oh it’s very interesting," said Alice. “I look at the grin and I see the eyes and whiskers and “I’ve heard something like that before," said Al- the ears and the warm furry body and the long ice crossly. “The creatures here all recite far too sinuous tail." much poetry." And she stalked angrily away. - Tutte, apparently apocryphal; found at http://userhome.brooklyn.cuny.edu/skingan/matroids/toast.html v Contents Approval ii Partial Copyright Licence iii Abstract iv Tutte on matroids v Table of Contents vi List of Figures ix Overview 1 1 Introduction 5 1.1 Matroids . 5 1.1.1 Matroid minors . 8 1.2 Biased graphs and frame matroids . 11 1.2.1 Group-labelled graphs . 11 1.2.2 Biased graphs represent frame matroids . 15 1.2.3 Minors of biased graphs . 19 1.2.4 Biased graph representations . 24 1.3 Some useful technical tools . 27 1.3.1 Rerouting . 27 1.3.2 A characterisation of signed graphs . 28 1.3.3 Biased graphs with a balancing vertex . 29 1.3.4 Pinches and roll ups . 31 1.3.5 Connectivity . 33 1.3.6 How to find a U2;4 minor . 34 2 When is a biased graph group-labellable? 38 2.1 Context and preliminaries . 39 2.1.1 Lift matroids . 40 vi 2.1.2 Branch decompositions . 42 2.1.3 Spikes and swirls . 43 2.2 A Topological Characterisation . 44 2.2.1 Group-labelling by arbitrary groups . 46 2.3 Constructing minor-minimal non-group-labellable biased graphs . 49 2.4 Excluded Minors — Biased Graphs . 51 2.5 Excluded Minors — Matroids . 57 2.5.1 Excluded minors — frame matroids . 57 2.5.2 Excluded minors — lift matroids . 60 2.6 Infinite antichains in GΓ, FΓ, LΓ ......................... 64 2.7 Finitely group-labelled graphs of bounded branch-width . 66 2.7.1 Linked branch decompositions and a lemma about trees . 67 2.7.2 Rooted Γ-labelled graphs . 68 2.7.3 Proof of Theorem 2.5 . 69 3 Biased graph representations of graphic matroids 72 3.1 Six families of biased graphs whose frame matroids are graphic . 72 3.2 Proof of Theorem 3.1 . 76 4 On excluded minors of connectivity 2 for the class of frame matroids 82 4.1 On connectivity . 84 4.1.1 Excluded minors are connected, simple and cosimple . 84 4.1.2 Separations in biased graphs and frame matroids . 85 4.2 2-sums of frame matroids and matroidals . 86 4.2.1 2-summing biased graphs . 87 4.2.2 Decomposing along a 2-separation . 87 4.2.3 Proof of Theorem 4.7 . 94 4.3 Excluded minors . 95 4.3.1 The excluded minors E0 ......................... 95 4.3.2 Other excluded minors of connectivity 2 . 96 4.3.3 Excluded minors for the class of frame matroidals . 98 4.4 Proof of Theorem 4.1 . 99 4.4.1 The excluded minors E1 . 101 4.4.2 Finding matroidal minors using configurations . 105 4.4.3 Proof of Lemma 4.26 . 109 4.5 Some excluded minors of connectivity 2 not in E . 116 5 Representations of frame matroids having a biased graph representation with a balancing vertex 121 vii 5.1 Introduction . 121 5.2 Preliminaries . 123 5.2.1 Cocircuits and hyperplanes in biased graphs . 123 5.2.2 Committed vertices . 126 5.2.3 H-reduction and H-enlargement . 129 5.3 Proof of Theorem 5.1 . 131 5.3.1 All but the balancing vertex are committed . 132 5.3.2 Ω has ≥ 2 uncommitted vertices . 133 5.4 Biased graphs representing reductions of Ω . 156 6 Outlook 194 Bibliography 197 viii List of Figures Figure 1 Some minor-closed classes of biased graphs and of matroids. 2 Figure 2 A twisted flip. 4 Figure 3 A twisted flip’s effect on a single lobe . 4 Figure 1.1 The 4-point line U2;4 .......................... 7 Figure 1.2 The Fano matroid F7........................... 7 Figure 1.3 A biased graph . 11 Figure 1.4 Graphs embedded on a surface give rise to biased graphs. 12 Figure 1.5 Biased graphs representing excluded minors for graphic matroids . 13 Figure 1.6 Extending M(G) by V provides a frame for E. 15 Figure 1.7 Q8 is frame, but not linear . 17 Figure 1.8 The Vamos matroid . 18 Figure 1.9 A twisted flip . 26 Figure 1.10 A roll-up: F (G; B) =∼ F (G0; B0) ..................... 32 Figure 1.11 The biased graphs representing U2;4. 34 Figure 1.12 Finding U2;4 (i) . 35 Figure 1.13 Finding U2;3 (ii) . 35 Figure 1.14 Finding U2;4 (iii) . 36 Figure 2.1 A 2-complex for the Higman group . 47 Figure 2.2 Constructing a biased graph from K. 48 Figure 2.3 A Higman group-labelled graph .