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On Excluded Minors and Biased Graph Representations of Frame Matroids 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 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 un- balanced. 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 representations 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 matroids by seeking to determine those excluded minors that are not 3-connected. We come close, determining a set of 18 particular excluded minors and drastically narrowing the search for E 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 Contents Approval ii Partial Copyright Licence iii Abstract iv Table of Contents v List of Figures viii Overview 1 1 Introduction 5 1.1 Matroids . 5 1.1.1 Matroid minors . 7 1.2 Biased graphs and frame matroids . 10 1.2.1 Group-labelled graphs . 12 1.2.2 Biased graphs represent frame matroids . 14 1.2.3 Minors of biased graphs . 19 1.2.4 Biased graph representations . 23 1.3 Some useful technical tools . 26 1.3.1 Rerouting . 27 1.3.2 A characterisation of signed graphs . 27 1.3.3 Biased graphs with a balancing vertex . 28 1.3.4 Pinches and roll ups . 30 1.3.5 Connectivity . 32 1.3.6 How to find a U2;4 minor . 33 2 When is a biased graph group-labellable? 36 2.1 Context and preliminaries . 37 2.1.1 Lift matroids . 38 2.1.2 Group-labelled graphs and matroid minors . 40 v 2.1.3 Branch decompositions . 40 2.1.4 Spikes and swirls . 41 2.2 A Topological Characterisation . 42 2.3 Constructing minor-minimal non-group-labellable biased graphs . 44 2.4 Excluded Minors | Biased Graphs . 47 2.5 Excluded Minors | Matroids . 52 2.5.1 Excluded minors | frame matroids . 52 2.5.2 Excluded minors | lift matroids . 55 2.6 Infinite antichains in , , ........................... 59 GΓ FΓ LΓ 2.7 Finitely group-labelled graphs of bounded branch-width . 61 2.7.1 Linked branch decompositions and a lemma about trees . 62 2.7.2 Rooted Γ-labelled graphs . 63 2.7.3 Proof of Theorem 2.5 . 63 3 Biased graph representations of graphic matroids 67 3.1 Six families of biased graphs whose frame matroids are graphic . 67 3.2 Proof of Theorem 3.1 . 71 4 On excluded minors of connectivity 2 for the class of frame matroids 74 4.1 On connectivity . 76 4.1.1 Excluded minors are connected, simple and cosimple . 76 4.1.2 Separations in biased graphs and frame matroids . 77 4.2 2-sums of frame matroids and matroidals . 78 4.2.1 2-summing biased graphs . 79 4.2.2 Decomposing along a 2-separation . 79 4.2.3 Proof of Theorem 4.7 . 85 4.3 Excluded minors . 86 4.3.1 The excluded minors ............................ 86 E0 4.3.2 Other excluded minors of connectivity 2 . 87 4.3.3 Excluded minors for the class of frame matroidals . 89 4.4 Proof of Theorem 4.1 . 91 4.4.1 The excluded minors ............................ 93 E1 4.4.2 Finding matroidal minors using configurations . 97 4.4.3 Proof of Lemma 4.26 . 100 5 Representations of frame matroids having a biased graph representation with a balancing vertex 109 5.1 Introduction . 109 5.2 Preliminaries . 112 vi 5.2.1 Cocircuits and hyperplanes in biased graphs . 112 5.2.2 Committed vertices . 114 5.2.3 H-reduction and H-enlargement . 117 5.3 Proof of Theorem 5.1 . 119 5.3.1 All but the balancing vertex are committed . 120 5.3.2 Ω has 2 uncommitted vertices . 121 ≥ 5.4 Biased graphs representing reductions of Ω . 143 6 Outlook 181 Bibliography 184 vii List of Figures Figure 1 Some minor-closed classes of biased graphs and of matroids. 2 Figure 2 A twisted flip. 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. 11 Figure 1.5 Biased graph representations of 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 F (G; ) = F (G0; 0).............................. 31 B ∼ B Figure 1.11 The biased graphs representing U2;4. .................... 33 Figure 1.12 Finding U2;4 (i) . 34 Figure 1.13 Finding U2;3 (ii) . 34 Figure 1.14 Finding U2;4 (iii) . 35 Figure 2.1 F8. ....................................... 47 Figure 2.2 H6........................................ 48 Figure 2.3 t-coloured planar graphs . 51 Figure 2.4 Modifying F2k ................................. 51 Figure 3.1 A curling . 68 Figure 3.2 A fat theta . 69 Figure 3.3 A 4-twisting . 70 Figure 3.4 A fat 7-wheel and a twisted fat 7-wheel . 71 Figure 4.1 A twisted flip . 75 Figure 4.2 Four types of biseparations. 80 Figure 4.3 Possible decompositions of Ω into the parts . 82 Figure 4.4 Finding a representation in which the biseparation if type 1. 84 viii Figure 4.5 If just xy and yz contain unbalanced cycles . 84 C C Figure 4.6 Circuits of F (Ω) meeting both sides of the 2-separation. 85 Figure 4.7 Finding a representation in which the biseparation if type 1 (ii) . 85 ∗ 0 Figure 4.8 M (K3;3).................................... 87 Figure 4.9 Excluded minors for the class of frame matroidals with L > 1. 92 j j Figure 4.10 Alternate representations of ;:::; . 94 M1 M7 Figure 4.11 W4........................................ 95 Figure 4.12 Any proper minor of W is e ; e -biased. 96 4 f 1 2g Figure 4.13 Configurations used to find ;:::; ................... 98 M0 M8 Figure 4.14 More configurations. 99 Figure 4.15 A twisted flip: F (Ω) ∼= F (Γ). 108 Figure 5.1 Biased graphs representing F (reH (Ω)). 110 Figure 5.2 Biased graphs representing F (Ω). 111 Figure 5.3 Ω is 3-connected, but F (Ω v) is disconnected. 113 − Figure 5.4 Possible representations of large balanced or pinched subgraphs . 116 Figure 5.5 Case (a)i. Ω and reH (Ω) . 122 Figure 5.6 Case (a)i. Two H-enlargements . 123 Figure 5.7 Case (a)ii. Ω and reH (Ω) .
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