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Split Graphs, Unigraphs, and Tyshkevich Decompositions

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Split Graphs, Unigraphs, and Tyshkevich Decompositions Split Graphs, Unigraphs, and Tyshkevich Decompositions Rebecca Whitman Submitted in Partial Fulfillment of the Prerequisite for Honors in the Wellesley College Department of Mathematics under advisor Ann Trenk May 2020 c 2020 Rebecca Whitman Abstract This thesis examines several topics connected to split graphs and unigraphs. We first study a partially ordered set representation of split graphs defined by [13], connecting properties of poset elements, called blocks, to those of split graphs and vice versa. We also find an equivalence relation between poset elements, equivalent to changing the leaf vertices of a vertex of highest degree in a split graph, and provide results for when equivalence classes are closed under various conditions. In the second part of the thesis, we investigate an inverse operation on split graphs, giving proofs for previously stated results and establishing new propositions about the relationship between the inverse operation and unigraphs, induced subgraphs, and hereditary classes. In the third part of the thesis, we use Tyshkevich de- composition [17] to give a structural characterization of the hereditary closure of the set of unigraphs. This characterization implies several results about the nature of forbidden induced subgraphs for the hereditary class. i Acknowledgements I cannot offer enough thanks to my thesis advisor and professor Ann Trenk, for her guidance, confidence in my work, and constructive criticism throughout this project. I have learned so much from her over the past two years about how to create and communicate mathematics. I also would like to thank Michael Barrus for his advice and for suggesting the problem that became Chapters 4 and 5 of this thesis. I am grateful to my mathematics professors Megan Kerr, Jonathan Tannenhauser, Karen Lange, Clara Chan, Ismar Voli´c,Helen Wang, and Andy Schultz for mentoring and challenging me and sharing their enthusiasm for the subject. Thank you as well to H´el`ene Bilis for graciously agreeing to serve as the honors visitor. I am fortunate and thankful to have friends who encourage my work, laugh with me, and have helped make my time at Wellesley incredible. Last but not least, I would like to thank my family for their love, kindness, and support. ii Preface This thesis studies split graphs in two different contexts: as elements of a poset in their own right and as the building blocks of a graph composition operation. The goal of Chapter 1 is to introduce the reader to the graph theoretic definitions, objects, and operations under consideration. We provide definitions for degree sequences, unigraphs, induced subgraphs, and hereditary classes. Chapter 1 also discusses split graphs, with the class's forbidden induced subgraph and degree sequence characterizations, and includes background results on split partitions and threshold graphs. The final section introduces the canonical decomposition of a graph as defined by Tyshkevich [17] (hereafter Tyshkevich decomposition). Chapter 2 presents a fourth characterization of split graphs. All degree sequences cor- responding to graphs can be reshaped into elements that form a poset under a majorization operation. The elements corresponding to split graphs take a recognizable, characterizable form [13]. The chapter also introduces an equivalence relation on these poset elements. We give results on the extent to which various properties of split graphs can be identified by the poset representations of split graphs, and are preserved within equivalence classes. In Chapter 3, we examine an inverse operation on split graphs that is used in [17] to identify all indecomposable unigraphs. Tyshkevich states a number of properties about the inverse, and we provide proofs for completeness. Additionally, we give results about the interaction of the inverse with split graph properties, induced subgraphs, and hereditary classes. The final two chapters together state and prove a structural characterization of the hereditary closure of the class of unigraphs, that is, the class of all graphs that are induced in some unigraph. We consider nine graphs and graph families of indecomposable unigraphs iii more closely in Chapter 4, where, family by family, we find all non-unigraphs induced in an indecomposable unigraph. These graphs, together with the indecomposable unigraphs, are the indecomposable components of graphs in the hereditary closure. Chapter 5 contains the statement and proof of this theorem. We also present a lemma regarding the Tyshkevich decomposition of induced subgraphs, and preliminary work towards a forbidden induced subgraph characterization of the class. iv Contents Abstract i Acknowledgements ii Preface iii 1 An Introduction to Split Graphs and Tyshkevich Decompositions 1 1.1 Graph Theory . .1 1.2 Split Graphs . .5 1.3 Tyshkevich Decompositions . .8 2 Posets of Split Graphs 12 2.1 Introduction . 12 2.2 Partition and Block Posets . 12 2.3 Split Graphs . 15 2.4 Block Similarity . 17 2.5 Graph Properties . 20 2.6 Poset Properties . 23 3 The Inverse Operation on Split Graphs 28 3.1 Introduction . 28 3.2 Inverses Depend on the KS-Partition . 29 3.3 Involutions . 31 3.4 Inverses of Tyshkevich Compositions . 32 v 3.5 Properties Preserved by the Inverse . 34 3.6 Hereditary Classes and the Inverse . 35 4 Non-Unigraphs Induced in Indecomposable Unigraphs 38 4.1 Introduction . 38 4.2 Non-Split Graphs . 39 4.2.1 The cycle C5 ............................... 39 4.2.2 The family mK2 ............................. 40 4.2.3 The family U2(m; n)........................... 40 4.2.4 The family U3(m)............................. 40 4.3 Split Graphs . 42 4.3.1 The graph K1 ............................... 43 4.3.2 The family S(p; q)............................ 43 4.3.3 The family S2(p1; q1; : : : ; pr; qr)..................... 44 4.3.4 The family S3(p; q1; q2).......................... 44 4.3.5 The family S4(p; q)............................ 48 5 A Structural Characterization of the Hereditary Closure of the Unigraphs 53 5.1 Introduction . 53 5.2 Characterization Theorem . 54 5.3 Properties of the Hereditary Closure . 57 6 Open Questions 60 References 62 vi Chapter 1 An Introduction to Split Graphs and Tyshkevich Decompositions 1.1 Graph Theory In the first section of this chapter, we give necessary background in graph theory for the topics covered in this thesis. For more information, consult Douglas West's Introduction to Graph Theory [18], a standard graph theory reference. Definition 1.1. A graph G is an ordered pair consisting of a vertex set V (G) and an edge set E(G) whose elements are unordered pairs of distinct vertices. For the purposes of this thesis, we will restrict our discussion to finite, simple graphs, that is, those whose vertex set is finite, and contain neither edges from a single vertex to itself nor multiple edges between the same pair of vertices. Figure 1.1 gives several examples of important small graphs. Of these graphs, the K in K1 and 2K2 refers to a clique (see Definition 1.11) on 1 or 2 vertices, the P in P4 designates a path (line), and C4 and C5 are cycles (polygons) of 4 and 5 vertices, respectively. Up to vertex labeling, two graphs are the \same" when they are isomorphic, which we define here. Definition 1.2. An isomorphism from graph G to graph H is a bijection f : V (G) ! V (H) such that uv 2 E(G) if and only if f(u)f(v) 2 E(H). If such a function exists, then we say that G and H are isomorphic, denoted G ∼= H. 1 Figure 1.1: Five graphs that appear throughout the thesis Isomorphisms preserve all graph properties except with respect to vertex labeling. In the remainder of the section, we will introduce definitions related to three types of graph properties: vertex degree, subgraphs, and graph complements. Definition 1.3. Two vertices u; v 2 V (G) are adjacent if uv 2 E(G). The neighbor set N(v) of a vertex v is the set of vertices adjacent to v. The degree of vertex v, denoted degG(v), is the size of its neighbor set N(v) in graph G. For example, all vertices in C5 have degree 2. In this thesis, we frequently compare the degree of a single vertex v across several graphs, so we are careful to specify the graph in question as a subscript. We extend the notions of vertex adjacencies and degree globally. Definition 1.4. A graph G is connected if for all pairs of vertices u; v, there exists a sequence of edges ux1; x1x2; : : : ; xnv 2 E(G), called a path, from u to v. Each of the graphs in Figure 1.1 is connected except 2K2. Definition 1.5. A degree sequence is a list of non-negative integers written in non-increasing order, as d1 ≥ d2 ≥ ::: ≥ dn. It is worthwhile to examine when a degree sequence is the set of vertex degrees of some graph. 2 Figure 1.2: Two non-unigraphs realizing the degree sequence 3; 2; 2; 2; 1; with an induced P4 subgraph in bold Definition 1.6. A degree sequence is graphic if its entries are the vertex degrees of some graph G, and we say that G realizes that degree sequence. Furthermore, a degree sequence is unigraphic if it realizes exactly one graph, namely G, up to isomorphism. If so, we call G a unigraph or say it is unigraphic. Otherwise, G is non-unigraphic. For example, the degree sequence 2; 2; 1; 1 is realized by the graph P4. Though the small graphs of Figure 1.1 are unigraphs, many graphs are non-unigraphic. Figure 1.2 gives an example of two non-isomorphic graphs that realize the degree sequence 3; 2; 2; 2; 1. Graphic sequences are characterized, and [18] includes the statement and proof of this result.
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