Jonathan Miller Graphs that are minor minimal not Apex of connectivity κ 6 5 Honors in the Major Thesis California State University Chico Department of Natural Sciences Supervision Prof. Dr. Thomas Mattman July 2016 Contents List of Figures iii List of Tables iv Nomenclature iv Abstract 1 1 Introduction 2 1.1 Basics . .2 1.1.1 Definitions . .2 1.1.2 Operations of Graphs . .3 1.2 Graph Minor Theorem . .5 1.2.1 Minors . .5 1.2.2 The Family of Apex Graphs . .6 1.2.3 Minor Minimal Not Apex . .7 1.3 Results . .7 1.3.1 K6 and Jørgensen graphs . .7 2 What do we know about MMNA graphs 9 2.1 Introduction . .9 2.1.1 Minor Minimal Intrinsically Knotted . .9 2.2 MMNA Graphs . 10 2.2.1 Disconnected MMNA graphs . 10 2.2.2 Connectivity of MMNA graphs . 10 2.3 Connectivity 5 and MMNAK6 and J ......................... 11 2.4 3−connected constructions . 13 2.5 Content to Come . 14 − 3 The root of connectivity κ = 5 MMNA is K6 15 3.1 Introduction . 15 3.1.1 Properties of MMNA .............................. 15 − 3.1.2 K6 ........................................ 15 3.2 Bollobás . 16 4 Constructing MMNAs 21 4.1 Graphs that are 3−connected . 21 4.2 Tutte . 21 4.3 Creating candidates for MMNA tests . 22 4.3.1 3−connected to 5−connected . 23 4.4 Results of graph construction upto 8 vertices. 23 4.4.1 jGj = 6 ...................................... 23 4.4.2 jGj = 7 ...................................... 23 i 4.4.3 jGj = 8 ...................................... 25 4.5 Conclusions . 27 5 Conclusion and Questions 28 5.1 Introduction . 28 − 5.1.1 Connecting K6 to MMNA or κ = 5 ...................... 28 5.1.2 Building κ = 5 graphs from 3−connected ones . 28 5.2 Questions remaining . 29 5.2.1 Proving K6 and Jørgensen are the only MMNA κ = 5 graphs . 29 A Definitions, Theorems, etc. 30 B To be completed 31 C Bibliography 33 Index 35 List of Figures 1.1 A simple graph on 5 vertices and 7 edges. .2 ∼ 1.2 Edge contraction K5=e = K4 ..............................4 1.3 Vertex split options of K4 ................................4 1.4 K6 and J .........................................7 2.1 The Petersen Family - creative commons . 10 3.1 The Icosahedral Graph, I12 ............................... 16 − + 3.2 K6 < I12 ......................................... 17 3.3 I12 ............................................. 18 3.4 I12 top........................................... 18 4.1 W5 and DW4 ....................................... 21 4.2 W4 ............................................. 24 4.3 Case a ........................................... 24 4.4 Case b ........................................... 24 iii List of Tables iv Nomenclature ρ A graph property E(G) Edge set of a graph G G A finite simple graph G+ A graph with an added edge G− Graph with a removed edge G1 < G2 Graph G1 is a minor of G2 MMNA A graph G that is NA and also Apex when actions G=e and G − e are applied to G NA A graph G that is Not Apex is G − v is not planar V (G) Vertex set of a graph G v Abstract We embark on an exploration of graphs which are closed under taking minors. That is to say, a graph G with a certain property p, also has that same property p after we take a minor. In particular, we consider the case for all graphs that are apex by considering the finite class of graphs that are minor minimal not apex. Recall that a graph, G is apex if G − v is planar for some v 2 V (G). For the consideration of this report we consider only graphs that have vertex connectivity equalling to five. Specifically is the conjecture that there exist only two such graphs that are minor minimally not apex which are also with connectivity κ = 5, namely K6 and the prism graph on 6 vertices plus two additional vertices connected to all but themselves, which is called the Jørgensen graph. 1 Chapter 1 Introduction In 2004 Neil Robertson and Paul Seymour published the article "Graph Minors.XX. Wagner’s conjecture, . ,in which they proved that for every infinite set of finite graphs, one of its members is isomorphic to a minor of another" [4]. Meaning, that for graphs closed under taking minors, one can always find two graphs such that one of them is a minor of the other. We delve into the basics of Graph Theory to refresh the essential definitions and theorems that have been referenced for the purpose of this research. I begun the researching this topic in the Summer of 2016, continued on into my senior year of my BSc with Professor Thomas Matman for the purpose of honors in the major. I endeavour to delineate effectively the nature of Graph Theory as it pertains to the Graph Minor Theorem.[2] 1.1 Basics We begin this discussion on graphs with the following essentials to any discussion in Graph Theory. Generally any textbook on Graph Theory will cover these topics. Take Graph Theory by Reinhard Deistel, see [2], for a complete look at graph theory. 1.1.1 Definitions We recall the essentail definition of a graph: Definition 1: Graphs A graph is a pair G = (V; E) of sets, with E ⊂ V 2; that is the elements of E are 2 element subsets of V . The v 2 V are called vertices and the e 2 E are called edges. We use the notation e := uv. If u = v then G contains loops and potentially double edges and is called a multi-graph. We restrict multi-graphs and only consider simple graphs containing no double edges or loops. Figure 1.1 shows an example of a simple graphs B C A E D Figure 1.1: A simple graph on 5 vertices and 7 edges. 2 Chapter 1. Introduction 3 Order and size For a graph G with n vertices and m edges, we say jGj = n and jjGjj = m, and jGj, jjGjj is called the size and order respectively. Incidence, Adjacency, Neighbours, Degree For any edge there are 2 ends, say u and v. We say that uv is incident upon u or v. Also, we say that u and v are adjacent. Suppose that u is adjacent to k vertices, then the neighbourhood of v is the list of all k vertices and we denote this list by N(v), and its length or size by jN(v)j. The degree of v = jN(v). The graph G where all vertices are adjacent to all other vertices in G is said to be complete and is denoted as Kn. We say the complete graphs on n vertices. Subgraphs Let H be a subgraph of G, where H and G are both graphs and we notate as H ⊂ G. Then V (H) ⊂ V (G) and E(H) ⊂ E(G). If V (H) = V (G), then H is a spanning subgraph. If our subgraph V (H) is an arbitrary subset of V (G), and E(H) consists of all edges of G whose ends are all in V (H), then H is an induced subgraph of G. 1.1.2 Operations of Graphs Let G be a graph, then a graph operation is defined in the following boxes. Definition 2: Vertices and Edges Vertices For a graph G, we can add or subtract vertices. • G − v, is the graph G without v and all edges incident upon v. • G + v, is the new graph G + v with a new vertex v added to the old graph G, that may or may not have added edges. Edges For a graph G, we can add or subtract edges. • G − e, is the graph G with an edge removed from the existing graph. • G+e, is the existing graph G, where two non-adjacent vertices are now adjacent after G + e. Definition 3: Edge Contractions Let x and y be two adjacent vertices in a graph G, then G=xy is the graph formed from smashing x and y into a new vertex, z. We note that jG=xyj 6 jGj = n − 1. Then this is called an edge contraction ∼ Figure 1.2 shows K5=v = K4, through an edge contraction. 4 1.1. Basics B B C A A C D E D ∼ Figure 1.2: Edge contraction K5=e = K4 Definition 4: Vertex Splits Let G be a graph and let z be a vertex on n neighbors. A vertex split splits z into two new vertices x and y. We write G ? v(a; b), where a = jN(x)j − 1 and b = jN(y)j − 1. Figure 1.3 shows a 4 different vertex splits. The top left is written as G?A(0; 3). The bottom right is written as G?A(3; 3), which is also isomporphic to K5. Not all vertex splits will be the inverse operation of a given edge contraction. B B A A E E C C D D B B A A E E C C D D Figure 1.3: Vertex split options of K4 Definition 5: Connectivity Let G be a connected graph, U be a subset of vertices in G, and jUj < κ. G is κ−connected if G − κ is connected Connectivity A graph is connectivity κ = k, when κ is largest integer for which G − k is κ−connected. Chapter 1. Introduction 5 1.2 Graph Minor Theorem 1.2.1 Minors Definition 6: Graph Minor Let a graph H be a minor of another graph G. Then at least one of the following is true: 1. G − v =∼ H 2.