Graph Immersions by Mahdieh Malekian M.Sc., Sharif University of Technology, 2010 B.Sc., Sharif University of Technology, 2006 Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Department of Mathematics Faculty of Science ©Mahdieh Malekian 2018 SIMON FRASER UNIVERSITY Fall 2018 Copyright in this work rests with the author. Please ensure that any reproduction or re-use is done in accordance with the relevant national copyright legislation. Approval Name: Mahdieh Malekian Degree: Doctor of Philosophy (Mathematics) Title: Graph Immersions Examining Committee: Chair: Paul Tupper Professor Matt DeVos Senior Supervisor Associate Professor Bojan Mohar Supervisor Professor Luis Goddyn Internal Examiner Professor Department of Mathematics Guoli Ding External Examiner Professor Department of Mathematics Louisiana State University Date Defended: November 27, 2018 ii Abstract One of the first prominent theorems in structural graph theory is the Kuratowski-Wagner theorem which characterizes planar graphs as those with no K3,3 or K5 minor. Numerous other classical theorems give precise descriptions of the class of graphs with no H-minor for numerous small graphs H. In particular, such classifications exist when H is W4, W5, Prism, K3,3, K5, Octahedron, and Cube. One of the most useful tools in establishing such results are splitter theorems which reduce a graph while preserving both connectivity and containment of a given minor. In this thesis we consider analogous problems for a different containment relation: immer- sion. Although immersion is a standard containment relation, prior to this thesis there were almost no precise structure theorems for forbidden immersions. The most prominent theo- rems in this direction give a rough description of graphs with no W4 immersion and those with no K3,3 or K5 immersion. Our main contributions include precise structure theorems for the class of graphs with no H-immersion when H is one of K4,W4, Prism, and K3,3. To assist in this exploration, we have also established two splitter theorems for graph immersions, one for 2k-edge-connected graphs, and another for 3-edge-connected and internally 4-edge-connected graphs. Keywords: immersion; weak immersion; edge-connectivity; splitter theorem; chain theo- rem; structural theorem iii Dedication To all whose presence is an inspiration for becoming a better human being; I am indebted to so many of you for being in my life. iv Acknowledgements Pleased as I am about the outcome of my PhD research, I am much more delighted to have met many wonderful people thanks to the time spent at SFU. I would like to thank Matt DeVos for many things in addition to his most invaluable guidance all throughout this research, teaching me so much, and his brilliance in Mathe- matics. Matt! Thank you for your contagious enthusiasm, and for your huge patience with me; for all your encouragements and for your most generous support; for surpassing my wildest imaginations about the best supervisor. Matt! Thank you for being the amazing human being you are! I would also like to thank Stefan Hannie whose precious help is traceable in this thesis. Thank you Stefan for teaching me how to code in Sagemath, and for so patiently spending many hours debugging my code. Also, thanks for making Jarin and I so proud of you about the third month of each semester!! Thank you for exuding kindness, patience, and openness. Thank you Sebastián for the lovely human being you are, for your inspiring kindness, humility, and helpfulness—bi taruf! Thank you Baris for many refreshing chats, and for the darvish you are! Thank you Ruiping for your inspiring free spirit. My time at SFU was coloured by happy moments created by many Iranian fellow stu- dents who most generously offered their friendship (and their food!)—thank you all for bringing a taste of home to SFU! There are many others who are due thanks, at SFU and elsewhere; to whom I am forever grateful for adding so much beauty to the world I know. From those, I would like to particularly thank my family, who have always showered me with their unconditional love and care. I would also like to thank many dear friends, most of whom I missed out on so much during the past few years; thank you all for being the great people you are! I have learned from each one of you. v Table of Contents Approval ii Abstract iii Dedication iv Acknowledgements v Table of Contents vi List of Figures ix 1 Introduction 1 1.1 Overview . 1 1.2 Basic definitions . 3 1.3 Structural theorems for forbidden immersions . 4 1.3.1 K4 immersion . 4 1.3.2 W4 immersion . 5 1.3.3 Prism immersion . 5 1.3.4 Immersion of Kuratowski graphs . 6 1.4 Splitter Theorems . 6 1.5 More known results on graph minors/immersions . 9 1.5.1 Graph minors . 9 1.5.2 Graph immersions . 10 1.6 The story of this research program . 10 2 Splitter Theorems for Graph Immersions 13 2.1 k-edge-connected graphs, k ≥ 4 even . 14 2.1.1 Preliminaries . 14 2.1.2 k-edge-cuts . 16 2.1.3 (k + 1)-edge-cuts . 18 2.1.4 Finishing the proof of Theorem 2.1 . 20 2.2 3-edge-connected, internally 4-edge-connected graphs . 21 vi 2.3 Implication for immersing Kuratowski graphs . 27 3 Immersion of Dm 31 3.1 Proof of Theorem 3.4 . 33 3.1.1 ‘if’ direction . 33 3.1.2 ‘only if’ direction . 34 3.2 Immersion of K4 ................................. 39 3.2.1 Immersion of K4 with two roots . 39 3.2.2 Immersion of K4 with up to one root . 40 4 Immersion of W4 43 4.1 Rooted W4 immersions . 45 4.1.1 Proof of the ‘if’ direction of Theorem 4.2 . 45 4.1.2 Proof of the ‘only if’ direction of Theorem 4.2 . 46 4.1.3 A note about the code . 50 4.2 Unrooted W4 immersions . 51 4.3 Additional tools . 55 4.3.1 Reductions . 55 4.3.2 Tree-width; Branch-width . 56 4.4 Graphs with arbitrary edge-connectivity without a W4 immersion . 58 5 Immersion of Prism 59 5.1 A corollary of Theorem 2.17 . 61 5.2 Proof of Theorem 5.1 . 63 6 Immersion of Eyeglasses 67 7 Immersion of K3,3 72 7.1 Helpful tools for proof of Theorem 7.1 . 74 7.2 Proof of Theorem 7.1 . 76 7.2.1 4-edge-cuts . 77 7.2.2 Computation . 78 7.2.3 Two local properties . 79 7.2.4 5-edge-cuts . 80 7.2.5 Finishing the proof . 82 7.3 Corollaries of Theorem 7.1 . 83 7.3.1 Graphs of arbitrary edge-connectivity excluding a K3,3 immersion . 83 7.3.2 Tree-width and branch-width of graphs without K3,3 immersion . 83 Bibliography 86 vii Appendices 90 Appendix A Code used in the proof of Theorem 7.1 90 Appendix B A second proof of Theorem 4.2 118 B.1 Connectivity I . 120 B.2 Four neighbours . 122 B.3 One step above an obstruction . 125 B.4 Connectivity II . 130 B.5 Proof of main theorem . 133 viii List of Figures Figure 1.1 Sausage reduction . 3 Figure 1.2 A graph with a (3, 3)-segmentation of width four . 4 Figure 2.1 Cuts δ(X0), δ(Y ) relative to each other . 23 Figure 2.2 At least one of X0c ∩ Y, X0c ∩ Y c consists of only one vertex . 24 Figure 2.3 Both X0c ∩ Y, X0c ∩ Y c consist of two vertices . 24 Figure 2.4 Both uw and vz are in interesting edge-cuts . 24 Figure 2.5 The last graphs in the sequence . 28 Figure 2.6 Octahedron . 30 Figure 3.1 Graph D2 ................................ 31 Figure 3.2 Immersions in G when dG∗ (yi) = 1 or dG∗ (xi) = 0 . 35 Figure 3.3 Possibilities for G∗ ........................... 35 Figure 3.4 Types of Paths . 35 Figure 4.1 . 44 Figure 4.2 Type 3 graphs after sausage reduction . 44 Figure 4.3 Type 4 graphs . 44 Figure 4.4 Non-cubic sausage reduced graphs without a W4 immersion . 52 Figure 5.1 Prism graph . 59 Figure 5.2 Graphs of type 1 . 60 Figure 5.3 Graphs of type 2 after sausage reduction . 60 Figure 5.4 Graphs of type 3 after sausage reduction . 60 Figure 5.5 Graphs of type 4 . 61 Figure 5.6 Graphs of type 5 . 61 Figure 6.1 Eyeglasses . 67 Figure 6.2 Obstructions to an immersion of Eyeglasses . 68 Figure 7.1 Type 2 graphs . 72 Figure 7.2 Graphs of Type 3 after sausage reduction . 73 Figure 7.3 Type 4 graphs . 73 ix Chapter 1 Introduction 1.1 Overview The study of containment relations between graphs is a central topic in structural graph theory. A variety of containment relations may be defined when studying graphs, some of these, such as minor and topological minor are quite well-explored. A graph H is a minor of another graph G if a subgraph of a graph G can be transformed to H by a series of edge contraction. Graph H is a topological minor of G if G contains a subdivision of H as a subgraph. Arguably, the first significant structural theorem about graph minors is the Kuratowski-Wagner theorem, which characterizes planar graphs. Theorem 1.1 (Kuratowski [27]; Wagner [47]). A graph is planar if and only if it does not contain K3,3 or K5 as a minor (subdivision).