Masaryk University Faculty of Informatics Case analysis of a splitter theorem Bachelor’s Thesis Matúš Hlaváčik Brno, Spring 2016 Replace this page with a copy of the official signed thesis assignment and the copy of the Statement of an Author. Declaration Hereby I declare that this paper is my original authorial work, which I have worked out on my own. All sources, references, and literature used or excerpted during elaboration of this work are properly cited and listed in complete reference to the due source. Matúš Hlaváčik Advisor: prof. RNDr. Petr Hliněný, Ph.D. i Acknowledgement I would like to express my deepest gratitude to my advisor, prof. RNDr. Petr Hliněný, Ph.D., for his patience, helpful comments and suggestions. I would also like to thank my family and friends for their support provided not only while working on this thesis, but during all my life. ii Abstract After many years of research C. Chun, D. Mayhew, and J. G. Oxley published a new splitter theorem which we could use in an algo- rithm for finding internally 4-connected graphs which can have planar emulators. The purpose is to limit the possible counter-examples to Fellow’s planar emulator conjecture. In this work, we explain everything needed for understanding this new splitter theorem (contains pieces of graph theory, matroids theory and specific structures of matroids in graphs). This splitter theorem is originally for internally 4-connected binary matroids and is used to generate minors, so in this thesis we describe the version of the theorem for graphs and we turn the theorem "inside out". This enables us to generate bigger graphs while preserving some minor-related properties which are needed for generating possible counter-examples. iii Keywords graph, matroid, finite planar cover, finite planar emulator, internally 4-connected, two disjoint k-graphs, splitter theorem iv Contents 1 Introduction 1 2 Basic definitions 2 2.1 Graphs .............................2 2.2 Matroids ...........................7 3 Introduction of the main problem 9 3.1 Planar covers .........................9 3.2 Planar emulators ....................... 10 3.3 Current state of the problem ................. 11 4 Importance of a splitter theorem 17 5 The new splitter theorem 20 5.1 Operations from Theorem 5.1 ................. 21 5.2 The duals of the operations .................. 22 6 Application of the splitter theorem 39 6.1 Non-useful steps (iv),(v) ................... 41 6.2 Algorithm ........................... 41 7 Conclusion 44 Bibliography 45 v List of Figures 2.1 Example of a graph. 2 2.2 Examples of basic classes of graphs (Kn, Pn, Cn, Km,n). 3 2.3 The minor minimal obstruction for the projective plane. [8, 9] 6 2.4 G and G* (dashed) [6]. 8 3.1 (a) Graph G. (b) Planar cover of G. (c) Planar emulator of G. 10 3.2 The graph G = K5 (left) and his planar cover (right) [10]. 12 3.3 An example of a graph having two disjoint k-graphs [4]. 13 3.4 The planar emulator (right) of K4,5 − 4K2 (left) by Rieck and Yamashita [17]. 14 3.5 The graphs G and F (top) and 1-, 2- and 3-expansion of G by F (bottom line). 15 4.1 The operations of splitting a vertex, triad addition and triangle explosion [4]. 18 4.2 Quadrangular, pentagonal and hexagonal extension [4]. 19 5.1 (a) An augmented 4-wheel. (b) The result of deleting the central-cocircuit of an augmented 4-wheel. 24 5.2 (a) A ladder structure (n = 2). (b) The result of a ladder-compression move. 24 5.3 (a) An open rotor chain. (b) The result of trimming an open rotor chain. 25 5.4 (a) A ladder structure. (b) A result of trimming a ladder structure. 25 5.5 (a) A ladder structure. (b) A result of trimming a ladder structure. 26 5.6 (a) An open quartic ladder. (b) The result of a mixed ladder move. 27 5.7 (a) An enhanced quartic ladder. (b) The result of an enhanced ladder move. 28 vi 5.8 (a) An enhanced quartic ladder. (b) The result of an enhanced ladder move. 29 5.9 (a) A bowtie ring. (b) The result of trimming a bowtie ring. 29 5.10 (a) The dual of an augmented 4-wheel. (b) The result of dual of deleting the central-cocircuit of an augmented 4-wheel. 30 5.11 (a) The dual of a ladder structure (n = 2). (b) The result of the dual of a ladder-compression move. 30 5.12 (a) The dual of an open rotor chain. (b) The result of the dual of trimming an open rotor chain. 30 5.13 (a) The dual of a ladder structure. (b) A result of the dual of trimming a ladder structure. 31 5.14 (a) The dual of a ladder structure. (b) A result of the dual of trimming a ladder structure. 32 5.15 (a) The dual of a ladder structure. (b) A result of the dual of trimming a ladder structure. 33 5.16 (a) The dual of an open quartic ladder. (b) The result of the dual of a mixed ladder move. 34 5.17 (a) The dual of an enhance quartic ladder. (b) A result of the dual of an enhanced ladder move. 35 5.18 (a) The dual of an enhance quartic ladder. (b) A result of the dual of an enhanced ladder move. 36 5.19 (a) The dual of an enhance quartic ladder. (b) A result of the dual of an enhanced ladder move. 37 5.20 (a) The dual of a bowtie ring. (b) A result of the dual of trimming a bowtie ring. 38 5.21 (a) The dual of a bowtie ring. (b) A result of the dual of trimming a bowtie ring. 38 6.1 (a) Planar embedding of graph K2,2,2. (b) Planar embedding of graph Q3 (the cube). 40 vii 1 Introduction A graph is a collection of nodes (vertices) and its connections (edges). Its representation is usually a drawing. We interpret nodes as points and a connection of two nodes as lines between two connected nodes (points). Our often asked question is if we are able to draw a graph without an intersecting of some lines. In other words, if a graph can be em- bedded in a plane. We are easily able to solve this question because of Kuratowski’s theorem (2.2) [1]. However, what will happen if we allow multiplicity of vertices, but with the condition that we have to preserve the local structure of the graph? This is the question which arises from two independent studies in the 1980s by S. Negami (planar cover) and M. Fellows (planar emulator). Their studies are different for example in the definition of "the local structure", but surprisingly, their main conjectures are same. By the time, there were published some partial results which moved Negami’s conjecture closer to the proof. Nevertheless, Fellows’s conjecture had been stuck for 20 years until the end of 2008, when there was published a new surprising result and Fellows’s conjecture was disproved. In this thesis, we are following the approach used by P. Hliněný in [2] and then by M. Derka in [3] and [4]. We are explaining theory needed to understand a new splitter theorem and also restate the the- orem for our purpose. We want to be able to generate all possible counter-examples to Fellow’s planar emulator conjecture. After this introduction follows chapter with basic definitions and theory which is needed to understand before you can read the rest of this work. Chapter 3 is an introduction of the main problem and its terms and also the historical development of the main problem. In Chapter 4 we are explaining what is a splitter theorem and why it is so important for our research. Chapter 5 is presenting the new splitter theorem with theory needed to fully understand and in Chapter 6 there is a sketch of an algorithm for generating all possible counter- examples to Theorem 3.6. 1 2 Basic definitions This chapter is about basic theory of graphs and matroids. More infor- mation about graph theory can be found in [2] and about matroids in [5–7]. 2.1 Graphs A simple undirected graph is a pair G = (V, E), where V is a vertex set and E is an edge set. Edge e is a pair of vertices fu, vg. The vertex set of graph G referred as V(G) and the edge set as E(G). An edge e between u and v (e joins u and v) is denoted by fu, vg or also only uv. When vertices are joined by an edge then they are adjacent or we call them neighbours. Also, we can say the vertices are ends of edge e and they are incident with e. Graphs are often represented as points (vertices) connected by curves (edges), for example Figure 2.1. x w u v e Figure 2.1: Example of a graph. We denote degree of a vertex v in a graph G by degG(v) – it equals to the number of edges in G incident with v. Now we present basic classes of graphs (examples in Figure 2.2). We say that a graph is k-regular if all its vertices have degree k. A vertex of degree 3 is called cubic vertex and 3-regular graph is cubic graph. 2 2. Basic definitions A graph for which every pair of vertices are connected by an edge is called complete graph (clique).