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 decomposition [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 Volić,Helen Wang, and Andy Schultz for mentoring and challenging me and sharing their enthusiasm for the subject. Thank you as well to Hélène 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 corresponding 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 ∈ E(G) if and only if f(u)f(v) ∈ 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 ∈ V (G) are adjacent if uv ∈ 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 ∈ 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. We now introduce another group of deﬁnitions surrounding graphs contained within other graphs.

Deﬁnition 1.7. A subgraph of G is a graph H such that V (H) ⊆ V (G) and E(H) ⊆ E(G). If H is a subgraph of G, then we say that G contains H.

Subgraphs are created by removing vertices and edges from G. Although every edge in H must have both its endpoints present in V (H), it is possible to exclude edges from G whose endpoints are included in V (H). We use the stricter deﬁnition of induced subgraph to rule out this behavior.

Deﬁnition 1.8. An induced subgraph H of graph G is a subgraph where u, v ∈ V (H) and uv ∈ E(G) implies uv ∈ E(H). If so, we say that H is induced in G, and that H is the subgraph of G induced by its vertex set, V (H).

3 In Figure 1.2, the subgraph P4 is induced by vertices {a, b, c, d}, in bold, but not by {b, c, d, e}, since we cannot omit edge be. Induced subgraphs are determined completely by the choice of vertex subset. From this, we describe sets of graphs closed under taking induced subgraphs.

Deﬁnition 1.9. A family G of graphs is hereditary if every induced subgraph of a graph in G is also a graph in G.

All hereditary classes can also be characterized by what they do not include, with a forbidden induced subgraph characterization. Here, a graph G is an element of a hereditary class if and only if no forbidden induced subgraph is induced in G. Since every graph not in the class must contain an induced subgraph outside of the class, a minimal set of forbidden induced subgraphs always exists, although it may be inﬁnite. Many important results in graph theory characterize a hereditary class by ﬁnding a minimal set of forbidden induced subgraphs. For example, the bipartite graphs are precisely those containing no induced cycles of odd length. We introduce the graph complement operation, which “reverses” which edges are present in a graph.

Deﬁnition 1.10. The complement G of a graph G is the graph with vertex set V (G) and with uv ∈ E(G) if and only if uv 6∈ E(G).

For example, 2K2 and C4, shown in 1.1, are complements of one another, and P4 =

P4. We state without proof the standard result that induced subgraphs commute with complementation.

Proposition 1.1. If H is an induced subgraph of G, then H is the subgraph of G induced by V (H).

We deﬁne two types of vertex subsets and induced subgraphs, based on adjacency.

Deﬁnition 1.11. A clique, denoted K, is a subset of V (G) whose elements are pairwise adjacent. In contrast, a stable set, denoted S, is a subset of V (G) whose elements are pairwise non-adjacent. The clique number ω(G) is the maximum size of a clique in G, and the independence number α(G) is the maximum size of a stable set in G.

4 Whereas a clique contains all edges, a stable set contains no edges. For example, both

the clique and stable set numbers of P4 are 2. Hence, the complement of a subgraph induced on a clique is a stable set, and vice versa. Cliques and stable sets are essential for deﬁning split graphs, which will be introduced in the next section.

1.2 Split Graphs

Földesand Hammer [10] first defined split graphs in 1977 and they have been studied ever since. As a subclass of perfect graphs, split graphs are presented in detail in Golumbic’s book Algorithmic Graph Theory and Perfect Graphs [11]. Updated results, including on balanced graphs, can be found in Collins and Trenk’s forthcoming chapter “Split Graphs” in Topics in Algorithmic Graph Theory [9]. The reader can consult either text for proofs to theorems and more information.

Deﬁnition 1.12. A graph G is a split graph if there exists a partition of the vertex set of G into a clique K and a stable set S. We call this partition a KS-partition.

A split graph may have multiple KS-partitions, with diﬀerent clique sizes. Figure 1.3 shows two split graphs, one of which is depicted twice, with diﬀerent KS-partitions. Since any induced subgraph of a split graph can inherit its KS-partition, the class of split graphs is hereditary. We provide its forbidden induced subgraph characterization.

Figure 1.3: Two split graphs; the solid vertices are the clique and the hollow vertices are the stable set of the given partitions.

5 Theorem 1.2. [10] A graph G is a split graph if and only if G contains no induced 2K2,

C4, or C5.

These three forbidden induced subgraphs are shown in Figure 1.1. The vertices in K are ﬁlled-in circles, whereas the vertices in S are hollow. Unlike the majority of graph classes, split graphs can also be identiﬁed from their degree sequences. We recount that theorem here.

Theorem 1.3. [12] A graph G on n vertices is split if and only if its degree sequence d =

(d1, d2, . . . , dn) satisﬁes the following equation for h = ω(G):

h n X X di = h(h − 1) + di i=1 i=h+1 Furthermore, it is possible to deduce the clique size in all possible KS-partitions of a split graph from its degree sequence, and so all split graphs realizing the same degree sequence have the same clique number. In [8], Collins and Trenk introduce terminology for split graphs with a unique KS-partition.

Deﬁnition 1.13. A split graph is balanced if there exists a unique partition of its vertices into a clique and a stable set, and unbalanced otherwise.

We can determine whether a split graph has one or multiple KS-partitions based on the presence of swing vertices, which we deﬁne here.

Deﬁnition 1.14. Given a split graph G and any KS-partition K ∪ S of G, a swing vertex v is a vertex of G whose neighbor set is exactly K − {v}.

Because swing vertices can “swing” back and forth between the clique and the stable set of a KS-partition, we give a remark about balanced graphs.

Remark 1.4. A split graph G is balanced if and only if it contains no swing vertices.

Furthermore, as a result of Hammer and Simeone’s work in [12], all KS-partitions of a split graph take one of three forms. We classify partitions by comparing |K| with ω(G), the clique number of G (see Deﬁnition 1.11), and |S| with α(G), the independence number. We call a KS-partition K-max when |K| = ω(G) and S-max when |S| = α(G); for an example, see Figure 1.3.

6 Theorem 1.5. Let G be a split graph with partition K ∪ S. Exactly one of the following holds:

1. |K| = ω(G) and |S| = α(G), in which case G is balanced.

2. |K| = ω(G) and |S| = α(G) − 1, in which case G is unbalanced and the partition is K-max.

3. |K| = ω(G) − 1 and |S| = α(G), in which case G is unbalanced and the partition is S-max.

Thus, a given split graph has at most two partitions, up to isomorphism. In addition to swing vertices, we assign names to vertices of particular degrees.

Deﬁnition 1.15. In any graph G, a vertex v ∈ V (G) is an isolated vertex if degG(v) = 0, a leaf vertex if degG(v) = 1, and a dominating vertex if degG(v) = |V (G)| − 1.

Hence, isolated vertices have no neighbors, leaf vertices (leaves) are adjacent to exactly one vertex, and dominating vertices are adjacent to every other vertex in G. For example, the top vertex of the left-hand graph in Figure 1.3 is a leaf vertex, and the single vertex in

K1 shown in Figure 1.1 is trivially both an isolated and dominating vertex. We conclude this introduction to split graphs with a discussion of one subclass of split graphs, the threshold graphs. Threshold graphs were first defined by Chvatáland Hammer [4] in 1973.

Deﬁnition 1.16. A graph G is a threshold graph if there exist a threshold t > 0 and a positive weight ai assigned to each vertex vi so that S ⊆ V (G) is a stable set if and only if P i∈S ai ≤ t.

An example of a threshold graph is given in Figure 1.4. There are many characterizations of threshold graphs, including the following. We will give further characterizations in Propositions 1.11 and 2.8.

Theorem 1.6. The following are equivalent for a graph G:

1. G is a threshold graph.

7 Figure 1.4: An example of a threshold graph

2. G contains no induced 2K2,C4, or P4.

3. G can be constructed from the empty graph by repeatedly adding an isolated vertex or a dominating vertex.

We omit the proof, but note that since P4 is an induced subgraph of C5, it follows that all threshold graphs are split graphs, by Theorem 1.2. The forbidden induced subgraphs of threshold graphs are shown in Figure 1.1. Additionally, threshold graphs are unbalanced by Remark 1.4, since the ﬁrst vertex v in the sequence of isolated and dominating vertices is a swing vertex. Every dominating vertex that is subsequently added is adjacent to v, and no isolated vertex added is adjacent to v. For example, in Figure 1.4, one possible vertex order to construct the graph is a (isolated/swing), c (isolated), b (dominating), and d (dominating). Split graphs are not necessarily unigraphic (see Figure 4.5 for a non-example), but threshold graphs are all unigraphic by the construction in Theorem 1.6. The next section, on a particular method of graph decomposition, includes an important result on identifying unigraphs.

1.3 Tyshkevich Decompositions

Across graph theory, numerous operations have been deﬁned to combine and separate graphs. Here, we introduce a relatively recent one that decomposes any graph into split graph components and one free component. This graph decomposition operation, ﬁrst introduced by Tyshkevich [16], [17], acts like factorization, where indecomposable split graphs stand in for

8 prime numbers, the fundamental building blocks. We deﬁne the composition of two graphs, then extend to n graphs and decomposition.

Deﬁnition 1.17. Let G be a split graph with KS-partition K ∪ S, and let H be any graph. Then, the composition of G and H, denoted G ◦ H, is the graph on V (G) ∪ V (H) with edge set E(G) ∪ E(H) ∪ {uv|u ∈ K, v ∈ V (H)}.

For example, consider G = P4 ◦ C4, where the clique vertices in the unique KS- partition of P4 are ﬁlled in:

Thus, G ◦ H is the disjoint union of the two graphs, together with all edges from the designated clique in G to vertices in H. While H may not be a split graph, the composition G ◦ H is split if and only if H is split. Changing the KS-partition of G always changes the composition, so in the case of unbalanced graphs, it is essential to specify the KS-partition of split graphs under composition. It is straightforward to extend the composition to n split graphs and one graph chosen freely. We deﬁne indecomposable graphs as those that cannot be the result of a non-trivial graph composition.

Deﬁnition 1.18. A graph G is decomposable if there exist non-empty graphs G1,G0, where

G1 is a split graph and G = G1 ◦ G0, and indecomposable otherwise.

We state the existence and uniqueness of Tyshkevich decomposition; a proof is found in [17].

Theorem 1.7. Every graph G can be written as a composition of indecomposable compo-

nents, where each of Gn,...,G1 has a ﬁxed KS-partition:

G = Gn ◦ ... ◦ G1 ◦ G0.

9 Furthermore, when each Gi is non-empty, this decomposition is unique up to isomorphism, where neither the order of the components nor the choice of partitions can vary.

Finding a graph’s unique Tyshkevich decomposition is not easily done on sight. There is a degree sequence characterization of indecomposable graphs in [17], but the characterization by Barrus and West [3] of indecomposable graphs and indecomposable components within a graph is much easier to use. We present that theorem in terms of induced subgraphs, rather than hypergraphs.

Theorem 1.8. Create a graph A4(G) with V (A4(G)) = V (G) and with edge uv if and only if

some P4,C4, or 2K2 induced in G contains both u and v. Then, a graph G is indecomposable

if and only if A4(G) is connected. More generally, the vertex sets of the components of the

decomposition are the vertex sets of the connected components of A4(G).

Figure 1.1 depicts the induced subgraphs P4, C4, and 2K2. We can say more about indecomposable graphs with respect to balance and KS-partitions.

Proposition 1.9. [9] A split graph G is unbalanced if and only if G = H ◦ K1 for some split graph H.

Proof. Let H be a split graph, so H ◦ K1 is also split. Since the single vertex k ∈ K1 is

adjacent to exactly the clique vertices of H in the composition, k is a swing vertex in H ◦K1.

By Remark 1.4, H ◦ K1 is unbalanced. For the converse, consider an unbalanced graph G with a K-max partition K ∪ S. Since G is unbalanced, it has a swing vertex k ∈ V (G), whose neighbor set NG(k) = K − {k}. Let H be the subgraph of G induced on V (G) − {k}, 0 0 0 0 with an inherited partition K ∪ S , so K = K − {k}. Thus NG(k) = K , so k is adjacent to exactly the clique part of H, and G can be decomposed as G = H ◦ K1.

Thus, swing vertices are represented in decompositions with K1 as the rightmost term. As a corollary to Proposition 1.9, indecomposable graphs cannot be unbalanced with more than one vertex, or they can be decomposed as above.

Proposition 1.10. If G is an indecomposable split graph, then G is balanced (and hence

has a unique KS-partition) or is isomorphic to K1.

10 The converse does not hold, and many balanced split graphs are decomposable. We can similarly show that isolated and dominating vertices are represented in decompositions

with K1 as the leftmost term, contingent on partition. Based on the construction of threshold graphs in Theorem 1.6, we derive the following corollary.

Corollary 1.11. A graph G is threshold if and only if every element of its Tyshkevich

decomposition is K1, with either partition.

Although the class of unigraphs is not hereditary and therefore has no forbidden induced subgraph characterization, it can be characterized via Tyshkevich decomposition. In addition to stating and proving the following theorem, Tyshkevich [17] provides a list of all indecomposable unigraphs, completing the characterization.

Theorem 1.12. A graph G is unigraphic if and only if the components of its unique decomposition are indecomposable unigraphs.

Equivalently, if a collection of split graphs G1,...,Gn and a graph G0 are all unigraphic, then so too is their composition Gn ◦ ... ◦ G0. The indecomposable unigraphs are the focus of Chapter 4. We now set aside Tyshkevich decomposition and turn to a focused discussion of split graphs and subclasses in Chapter 2.

11 Chapter 2

Posets of Split Graphs

2.1 Introduction

Here, we consider a partially ordered set whose elements are ordered pairs of partitions of n into distinct integers. Each element of this poset will correspond to a degree sequence of a split graph. We will investigate the poset representation of split graph properties, and the graph representation of poset elements’ properties. For additional results and proofs, Merris’ Graph Theory [13] is an excellent resource.

2.2 Partition and Block Posets

We begin by constructing the well-known poset of partitions of a natural number into distinct parts. Let α = α1α2 . . . αm be a partition of n ∈ N into m distinct parts, such that α1 >

... > αm. Denote the length of α by len(α). As an example, consider α = 321, which is a partition of 6 into distinct parts, with len(α) = 3. We will continue this example throughout this chapter.

Deﬁnition 2.1. Denote the set of partitions of n ∈ N into distinct parts by Dis(n). Let Dis(n, m) be the (induced) subposet of Dis(n) comprising elements of length m.

Thus, α = 321 is an element of Dis(6) and in fact is the unique element of Dis(6, 3). We deﬁne an ordering on elements of Dis(n).

12 Deﬁnition 2.2. If α and β are two partitions of n into distinct parts, then β majorizes α, denoted by β α, when for all k ≤ len(β),

k k X X βi ≥ αi. i=1 i=1 Continuing our example, β = 42 is a partition of 6 into distinct parts, and 42 321. For k = 1, we have 4 > 3, and for k = 2, the sums of the ﬁrst two terms of β and α yield that 6 > 5. We provide a proof that the majorization operation induces a poset.

Proposition 2.1. The set Dis(n) is a poset ordered by majorization.

Proof. It is easy to see that for all α ∈ Dis(n), α α. The relation is anti-symmetric, since Pk Pk whenever β 6= α, there exists k ≤ len(β) with i=1 βi > i=1 αi. Since the sum of the ﬁrst k terms of α is strictly less than that of β, it follows that α 6 β. Lastly, majorization is transitive since the relation ≥ is transitive. Hence Dis(n) is a poset under majorization.

We can construct Hasse diagrams of these posets; for example, the Hasse diagram of Dis(6) is shown in Figure 2.1.

321

Figure 2.1: The Hasse diagram of Dis(6)

Next, we combine elements of Dis(n) into ordered pairs, called blocks.

Deﬁnition 2.3. A block is an ordered pair [α|β] where α, β ∈ Dis(n), β α, and len(β) = len(α) or len(β) = len(α) − 1. Let Block(n) denote the set of blocks with α, β ∈ Dis(n).

For example, [321|42] is a block, because 42 and 321 are each distinct partitions of 6, 42 321, and len(42) = len(321) − 1. The length condition in the block deﬁnition is

13 inspired by work in [14] and allows us to restrict our deﬁnition to blocks of split graphs, as we will see in Proposition 2.5. We next deﬁne what it means for one block to majorize another, and create a poset of blocks of partitions of n into distinct parts. Note that two types of majorization occur simultaneously, both within and between blocks.

Deﬁnition 2.4. If [α|β] and [α0|β0] are blocks with α, β, α0, and β0 ∈ Dis(n), then [α0|β0] majorizes [α|β] when β β0 and α0 α.

For example, [42|42] [321|42]: we compare the α terms and 42 321. Again, we conﬁrm that this majorization operation deﬁnes a poset.

Proposition 2.2. The set Block(n) is a poset ordered by block majorization.

Proof. Let [α|β], [α0|β0], and [α00|β00] ∈ Block(n). Since α α and β β, it follows that block majorization is reﬂexive. Suppose that [α0|β0] [α|β] and [α0|β0] 6= [α|β]. Thus, β β0 and α0 α. Both of these conditions are antisymmetric, so it follows that whenever β 6= β0, then β0 6 β, and when α 6= α0, then α 6 α0. Since the two blocks are not equal, we conclude that [α|β] 6 [α0|β0]. Suppose that [α00|β00] [α0|β0] [α|β]. Thus β β0 β00 and α00 α0 α. By transitivity of partition majorization, it follows that [α00|β00] [α|β]. Hence Block(n) is a poset ordered by majorization.

The Hasse diagram of Block(6) is shown in Figure 2.2.

[321|321][42|42] [51|51] [6|6]

[321|42] [42|51] [51|6]

[321|51] [42|6]

Figure 2.2: The Hasse diagram of Block(6)

14 2.3 Split Graphs

Now that we have shown Block(n) is a poset, we will argue that its elements correspond to degree sequences of split graphs. To begin, we consider a combinatorial tool, the Ferrers diagram, which can be used to represent degree sequences.

Deﬁnition 2.5. A Ferrers diagram is an array of boxes in left-justiﬁed rows. Rows are counted from the top, and the number of boxes in each row is non-increasing.

In a Ferrers diagram F , let αi be the number of boxes in the ith row starting at the

main diagonal, and βj be the number of boxes in the jth column starting below the main

diagonal. Figure 2.3 shows a Ferrers diagram with each αi and βj labeled. In [13], Merris

showed that each block is uniquely represented with a Ferrers diagram when {βj} majorizes

{αi}. We state this proposition and give a sketch of the proof. For more details, see [13].

Figure 2.3: The Ferrers diagram of block [321|42]

Proposition 2.3. There is a bijection between Block(n) and the set of Ferrers diagrams of

2n boxes where the sequence {βj} of column lengths below the main diagonal majorizes the

sequence {αi} of row lengths to the right of and including the main diagonal.

Proof. Let F be a Ferrers diagram under the given conditions. For the majorization condition

from Deﬁnition 2.2 to hold, α = {αi} and β = {βj} must partition the same integer; since the Ferrers diagram has 2n boxes, this integer is n. It can be shown that α and β are each distinct partitions of n if and only if the Ferrers diagram has non-increasing row lengths. Lastly, F has no blocks missing in the middle of its rows and has non-increasing row lengths

15 exactly when len(β) = len(α) or len(β) = len(α) − 1. We conclude that α, β ∈ Dis(n) and F corresponds uniquely to the block [α|β].

For instance, the Ferrers diagram of [321|42] is shown in Figure 2.3. There is a straightforward correspondence between a Ferrers diagram and a degree sequence d1, . . . , dt.

Namely, di is equal to the number of boxes in row i. Ruch and Gutman [15] show when such a degree sequence is graphic.

Proposition 2.4. The degree sequence d = d1, . . . , dn is graphic exactly when the sequences

{βj} and {αi} of its Ferrers diagram fulﬁll a weaker form of majorization, called weak sub- majorization.

Under weak sub-majorization, it is also possible to study larger posets of blocks corresponding to all graphs, not just split graphs. Both Merris [13] and Collins and Trenk [9] provide deﬁnitions of weak sub-majorization and are useful resources on these larger posets. For example, from the Ferrers diagram for [321|42] in Figure 2.3, we derive the degree sequence (3, 3, 3, 2, 1). The graph shown in Figure 2.4 is the unique graph realizing this degree sequence.

Figure 2.4: The split graph associated to the block [321|42]

Thus far, we have shown that blocks under majorization correspond to both Ferrers diagrams and degree sequences under some conditions. Our next result is that the elements of these posets correspond to split graphs, giving an additional perspective from which to study split graphs. We state the proposition from [13] and give a sketch of the proof.

16 Proposition 2.5. Each block [α|β] corresponds uniquely to a degree sequence of a split graph.

Proof. The proof is in two steps: ﬁrst, we must show that the degree sequence of [α|β] is graphic. This follows from Proposition 2.4 since β α implies that β weakly majorizes α. To verify that this graphic degree sequence indeed gives rise to split graph(s), we apply Theorem 1.3, the degree sequence characterization of split graphs. The calculation details can be found in multiple sources, including in [9].

There could be several split graph realizations of the same degree sequence, and consequently the same Ferrers diagram. We introduce notation to count these.

Deﬁnition 2.6. The realization number r([α|β]) is the number of non-isomorphic graphs arising from the degree sequence associated with block [α|β].

Thus, a block and its associated degree sequence are unigraphic if r([α|β]) = 1. For example, the realization number r([321|42]) is 1, since the three vertices of degree three must be adjacent to one another, and then each has exactly one more adjacency to one of the vertices of degrees one or two. Consider a second example: let α0 = 521 and β0 = 62, from Dis(8). Then [521|62] ∈ Block(8), and its Ferrers diagram is shown in Figure 2.5. We derive the degree sequence 5, 3, 3, 2, 1, 1, 1 from the row lengths of the Ferrers diagram. Since two non-isomorphic graphs, given in Figure 2.5 realize this degree sequence, it follows that r([521|62]) ≥ 2.

2.4 Block Similarity

In order to trace properties between elements of Block(n) for different natural numbers n, we define a notion of similarity between blocks, which forms an equivalence relation. Our equivalence relation will preserve many properties of both blocks and split graphs. As with the definitions above, we will begin at the level of partitions and extrapolate to blocks. We will then consider numerous properties of blocks or split graphs, establish parallel definitions in terms of the other category, and show, except in the cases of unigraphs and minimal blocks, that the properties are invariant among graphs in a shared equivalence class.

17 Figure 2.5: The two non-isomorphic split graphs whose degree sequence corresponds to the block [521|62].

Deﬁnition 2.7. Partition α0 ∈ Dis(n) is a shift of α ∈ Dis(m) by the integer n − m if

0 0 0 α1 − α1 = n − m and αk = αk for all 2 ≤ k ≤ len(α) = len(α ).

Thus, α and α0 diﬀer only in their ﬁrst element.

Deﬁnition 2.8. Let [α|β] ∈ Block(m) and [α0|β0] ∈ Block(n), with len(α) = len(α0) and len(β) = len(β0). These blocks are similar, denoted by [α|β] ∼ [α0|β0], if α0 is a shift of α by some integer k, and β0 is a shift of β by the same integer k.

For example, [321|42] ∼ [521|62], since 521 is a shift of 321 by 2, and 62 is a shift of 42 also by 2. We can also describe block similarity in terms of the graphs realized from each block: when two blocks are similar, each graph realizing one block differs from a graph realizing the other block only by the number of leaves adjacent to a specified vertex of highest degree. In other words, if [α|β] ∼ [α0|β0] by a shift of k > 0, we can take a graph G whose degree sequence corresponds to [α|β] and attach k additional leaves to a maximum degree vertex to get a graph G0 that realizes [α0|β0]. For example, Figure 2.4 shows a graph G corresponding to the block [321|42], and Figure 2.5 shows two graphs H and H0 which realize the block [521|62]. Observe that G can be obtained from H and H0 by removing 2 leaves from a maximum degree vertex, and H and H0 can each be obtained by adding 2 leaves to a maximum degree vertex of G. Additionally, if two graphs differ only by the number of leaves adjacent to a highest degree vertex, then their blocks are similar.

18 We now show that these shifts induce an equivalence relation.

Proposition 2.6. Block similarity is an equivalence relation.

0 0 Proof. Since α1 = α1 + 0 and β1 = β1 + 0, it follows that block similarity is reﬂexive. Let [α|β] ∼ [α0|β0] by a shift of k. Then [α0|β0] ∼ [α|β] by a shift of −k. To show transitivity,

0 0 00 00 0 0 00 0 let [α |β ] ∼ [α |β ] by a shift of l. Then α1 = α1 + k and α1 = α1 + l, and β1 = β1 + k and 0 00 00 00 00 00 β1 = β1 +l. Hence α1 = α1 +l+k and β1 = β1 +l+k, so we conclude that [α|β] ∼ [α |β ].

Since we can shift among similar blocks, we will focus on those in smallest possible form, which are useful for counting graph realizations.

Deﬁnition 2.9. A partition α ∈ Dis(n) is in root form if α1 = α2 + 1.

Deﬁnition 2.10. We similarly deﬁne a block [α|β] ∈ Block(n) to be in root form if either

α1 = α2 + 1 or β1 = β2 + 1.

This is the smallest possible form since we cannot shift α or β by a negative integer, to make their entries smaller. If we were to do so, then the entries of α or β would no longer be distinct and in strictly decreasing order. For example, α = 321 is in root form, but (42), (521), and (62) are not in root form. Since one of its partitions is in root form, [321|42] is in root form. The block [521|62] is not in root form, but it is similar to the root form block [321|42]. Root form blocks are of interest because they are the canonical representatives of their equivalence classes. Additionally, only graphs realizing root form blocks may have two or more vertices of highest degree. For instance, in Figure 2.4, vertices a and b are both of highest degree (3). Since no automorphism of the graph realizing [321|42] maps a to b, there are two non-isomorphic realizations of [321|42] shifted by 1, that is, the block [421|52], since we can append a leaf vertex to either a or b. Should we wish to shift [421|52] up 1 to [521|62], there are still exactly two non-isomorphic realizations of [521|62] since each of the two graphs realizing [421|52] has a unique vertex of highest degree. We summarize this process in the following proposition.

19 0 0 0 Proof. Shifting [α|β] to [α |β ] is equivalent to appending k = α1 −α1 leaf vertices to a vertex of highest degree in each realization of [α|β]. Let r([α|β]) = r and let G1,...,Gr be the r 0 non-isomorphic realizations of [α|β]. From each graph Gi, create graph Gi by identifying a vertex v of highest degree and append k leaves to v. Since Gi and Gj are pairwise non- 0 0 0 0 isomorphic, so too are Gi and Gj. Thus, G1,...,Gr are r non-isomorphic realizations of [α0|β0], so r([α0|β0]) ≥ r. When [α|β] is not in root form, it has already been shifted at least once from a root form block, so all graphs realizing [α|β] have a unique vertex of highest degree. Therefore,

0 each graph Gi has exactly one graph Gi corresponding to it. We can also remove k leaf ver- 0 0 0 0 tices from any graph realizing [α |β ] to get a graph realizing [α|β]. Therefore, {G1,...,Gr} is the set of graphs realizing [α0|β0], so r([α0|β0]) = r.

We conclude this section with several open questions on the relationship between similarity and the realization number. First and foremost, is it possible to deduce r([α|β]) from α and β from the block? We also hope to improve Proposition 2.7 and give an exact count of the realizations of [α0|β0], given a similar [α|β] in root form. Both are included on our list of open questions in Chapter 6.

2.5 Graph Properties

We now have the tools to address various properties of split graphs and connect them to the block representation of a graph. Throughout this section, we will refer to blocks, sequences, and graphs equally with properties normally attributed to graphs (e.g., a block is balanced when...) We begin with threshold graphs and state a result from Ruch and Gutman [15], then describe the eﬀect of similarity on threshold graphs.

Proposition 2.8. A block [α|β] corresponds to threshold graphs if and only if α = β.

Proposition 2.9. Similar blocks both correspond to threshold graphs, or neither does.

Proof. Let [α|β] be a threshold graph, so α = β. We shift [α|β] by k with the result that

0 0 0 0 α = α1 + k, α2, . . . , αm = β , so [α |β ] is also a threshold graph by Proposition 2.8.

20 The other direction is similar.

Figure 2.6 shows the Ferrers diagram and threshold graph corresponding to [32|32].

Figure 2.6: The Ferrers diagram and threshold graph realizing block [32|32]

Recall that a split graph is balanced if there exists a unique partition of its vertices into a clique and a stable set, and unbalanced otherwise. We state a characterization of balanced split graphs in terms of blocks that appears in [8].

Proposition 2.10. A block [α|β] is unbalanced if and only if len(α) = len(β), and conse- quentially, [α|β] is balanced exactly when len(α) = len(β) + 1.

Since shifting does not change the length of either α or β, similarity preserves balance, which we state as a corollary.

Corollary 2.11. Similar blocks are both balanced or both unbalanced.

Another type of graphs, Nordhaus-Gaddum graphs, generalize split graphs. They were initially characterized by Collins and Trenk in [7], through limitations on the chromatic number of the graph and its complement. Further work by Cheng, Collins, and Trenk, including applications to split graphs, is found in [6]. Nordhaus-Gaddum graphs are the graphs whose vertices can be partitioned into a clique (B), a stable set (C), and a nonempty third set (A) that is either a clique, a stable set, or C5. All possible edges between A and B are present, and there are no edges between A and C. Based on the options for A, there are three types of Nordhaus-Gaddum graphs.

21 Deﬁnition 2.11. A graph is an NG-1 graph if A is a clique, NG-2 graph if A is a stable

set, and NG-3 graph if A is C5. A graph is both NG-1 and NG-2 if and only if A is a single vertex.

We summarize results about split graphs from [6]. If a graph G is NG−1, then A ∪ B is a clique and C a stable set, so G is a split graph. If G is NG-2, then A ∪ C is a stable set,

and B a clique, so G is again a split graph. Since C5 is a forbidden induced subgraph of split graphs, it follows that NG-3 graphs are not split graphs. In both NG-1 and NG-2 graphs, all vertices of A are swing vertices. Since A is nonempty, these graphs are unbalanced. We will connect blocks to Nordhaus-Gaddum graphs via another characterization of the latter.

Let d1 ≥ d2 ≥ ... ≥ dn be a degree sequence satisfying the split graph criterion in Theorem 1.3, and let h be the value deﬁned in Theorem 1.3. The degree sequence corresponds to an

NG-1 graph if dh > dh+1 and to an NG-2 graph if dh−1 > dh. Should both conditions occur, the graph is both NG-1 and NG-2. For instance, the graph in Figure 1.4 which realizes the block [32|32] has degree sequence 3, 3, 2, 2, with h = 3, so this graph is NG-2. It may be helpful to refer to this example when reading the proof of Proposition 2.12 for NG-2 graphs. We classify unbalanced blocks as NG-1 or NG-2 based on the ﬁnal elements of α and β.

Proposition 2.12. Let [α|β] be a block and let len(α) = len(β) = m. Then [α|β] is NG-1

if and only if βm = 1, and [α|β] is NG-2 if and only if αm ≥ 2. Furthermore, all unbalanced split graphs exhibit one or both of these properties.

Proof. Consider the unique degree sequence d corresponding to [α|β], and ﬁx h as deﬁned

in Theorem 1.3. Since [α|β] is unbalanced by Proposition 2.10, it is known that dh = dm+1,

and that the vertex with degree dh will be a swing vertex, with m squares in its row of the

Ferrers diagram. Then [α|β] has βm = 1 if and only if dh+1 < dh, since the single Ferrers

diagram square implied by βm = 1 occurs in row h, which is equivalent to [α|β] being NG-1.

On the other hand, [α|β] has αm ≥ 2 exactly when dh−1 contains h − 2 + αm ≥ h = m + 1

squares, but dh has exactly m squares. Hence dh−1 > dh and [α|β] is NG-2.

For a contradiction, if neither property occurs for a block [α|β], then bm > am and so Pm−1 Pm−1 i=1 βi < i=1 αi and β 6 α.

22 We note further that if these properties occur simultaneously, then [α|β] is both NG-

1 and NG-2. Since these conditions are irrespective of α1 and β1, they are invariant under similarity. We state this as a corollary.

Corollary 2.13. For any two similar blocks, both correspond to NG-1 graphs or neither does, and both correspond to NG-2 graphs or neither does.

2.6 Poset Properties

We now return to the posets Dis(n) and Block(n), and translate poset properties to the generated split graphs. More precisely, the minimum and maximum elements of Dis(n, m) will give rise to elements of Block(n) and classes of split graphs with certain properties, including a hereditary class. We begin with results about maximum elements, which are easier to prove, then proceed to minimum elements. The following result is well-known; we provide a proof for completeness.

m Proposition 2.14. In Dis(n, m), an element β is maximum if and only if β1 = n − 2

and βi = m + 1 − i for 2 ≤ i ≤ m.

Proof. Let α ∈ Dis(n, m). Thus αi ≥ m + 1 − i for all i since α is a partition into strictly

decreasing parts. Then αi ≥ βi for all 2 ≤ i ≤ m, and denote the diﬀerence by ki = αi − βi. Thus m m m m X X X X α1 = n − αi = n − βi − ki = β1 − ki. i=2 i=2 i=2 i=2 Let 1 ≤ s ≤ m, and we will show that β α: s s m s s s m s X X X X X X X X αi = α1 + αi = β1 − ki + βi + ki = βi − ki ≤ βi. i=1 i=2 i=2 i=2 i=2 i=1 i=s+1 i=1 Therefore β α for all α ∈ Dis(n, m). Since majorization is anti-symmetric, we conclude that β is a unique maximum element in Dis(n, m). The reverse direction holds since every poset Dis(n, m) contains an element β with the given restrictions.

For instance, in Dis(6, 2) from Figure 2.1, the partition 51 is maximum, and in Dis(6, 3), the partition 321 is (trivially) maximum. Maximum elements of Dis(n, m) correspond to hairy split graphs, which we deﬁne here.

23 Deﬁnition 2.12. A split graph is hairy when there exists a partition of its vertices into a clique and a stable set where all vertices in the stable set have degree 0 or 1.

These graphs are called “hairy” since when drawn with the clique at the center, the leaves emanating from clique vertices look like whiskers or hairs. For example, both graphs of Figure 1.3 are hairy. It is only necessary for one KS-partition to satisfy the criteria, and for the unbalanced graph in Figure 1.3, the K-max partition satisﬁes the criteria. Tyshkevich [17] has shown that hairy split graphs are unigraphic, and we examine the class of indecomposable hairy graphs in Section 4.3.3. Furthermore, we can show that the class of hairy split graphs is hereditary, and that its forbidden induced subgraph characterization forbids the graph realized by [32|32] (see Figure 1.4) though we omit the proof in the interest of cohesion.

Proposition 2.15. A block [α|β] corresponds to a hairy split graph if and only if β is maximum in Dis(n, m).

Proof. Let [α|β] ∈ Block(n) with β maximum in Dis(n, m), and ﬁx h as deﬁned in Theorem

1.3. Then, βi = m + 1 − i for all 2 ≤ i ≤ m if and only if all columns of the Ferrers diagram

but β1 have their lowest square in row h, which is the ﬁnal row corresponding to clique vertices in a K-max partition of any graph of [α|β]. Lastly, the row h + 1 and all subsequent

rows have at most one square, from β1, if and only if all vertices of a minimal stable set in a split partition have degree 1.

For example, in Block(6), shown in Figure 2.2, the elements [51|51], [42|51], and [321|51] all correspond to hairy split graphs, since β = 51 is maximum in Dis(6, 2). It is notable that maximum values of β correspond to the hereditary class of hairy graphs, since no other combination of maximum or minimum α and/or β yields a hereditary class.

Since the condition for β to be maximum is irrespective of β1, the next corollary follows from Proposition 2.15.

Corollary 2.16. Similarity preserves whether graphs are hairy: two similar blocks are either both hairy or neither is.

24 We now consider minimum elements in Dis(n, m) and minimal blocks. For example, the partition 42 is minimum in Dis(6, 2) from Figure 2.1, and the partition 321 is (trivially) minimum in Dis(6, 3). From the Hasse diagram of Block(6) in Figure 2.2, a minimal block is [321|51], which is realized by the leftmost graph in Figure 1.3.

Proposition 2.17. In Dis(n, m), an element α is minimum when α1 − αm ≤ m.

Proof. Let α ∈ Dis(n, m). Since the terms of α are distinct, it holds that α1 − αm ≥ m − 1.

Let β ∈ Dis(n, m), and let βi − αi = si for all 1 ≤ i ≤ m. We separate into two cases:

In case 1, α1 − αm = m − 1, so each term of α is one smaller than the previous

term. Here, the si terms are non-increasing, because si − si+1 = βi − αi − βi+1 + αi+1 =

βi − βi+1 − 1 ≥ 0. Thus, for all 1 ≤ t ≤ m, t m m m X X X X si ≥ si = β − α = 0. i=1 i=1 i=1 i=1 We thus conclude that t t t t X X X X βi = si + αi ≥ αi, i=1 i=1 i=1 i=1 so β α.

In case 2, α1 − αm = m, so there exists some term αk with αk − αk+1 = 2 for some 1 ≤ k ≤ m − 1, and all other consecutive diﬀerences are one. By the argument in case 1, the

si terms for 1 ≤ i ≤ k are non-decreasing, as are the si terms from k +1 to m. Because there

is only one gap between the terms of α, it follows that either βi ≥ αi for i ≤ k, and hence Pk Pm Pm i=1 si ≥ 0, or βi ≤ αi for i ≥ k + 1, and hence i=k+1 si ≤ 0. Since i=1 si = 0 as β and Pk α both partition n, it follows that each partial sum implies the other, and that i=1 si ≥ 0. Therefore, if 1 ≤ t ≤ k, t k X X si ≥ si ≥ 0, i=1 i=1 and for k + 1 ≤ t ≤ m,

t m t t k m X X X X X X si ≥ si, so si = si + si ≥ si = 0. i=k+1 i=k+1 i=1 i=k+1 i=1 i=1

Therefore for all 1 ≤ t ≤ m, the sum of the si terms is non-negative, so we can conclude that t t t t X X X X βi = si + αi ≥ αi, i=1 i=1 i=1 i=1

25 and β α. Therefore β α for all β ∈ Dis(n, m). Since majorization is anti-symmetric, we conclude that α is a unique minimum element in Dis(n, m).

We add a lemma about majorization within Dis(n), which will then allow us to characterize the minimal elements of Block(n) in Proposition 2.19.

Lemma 2.18. If α, β ∈ Dis(n) and len(α) > len(β), then α 6 β.

Proof. Consider the partial sum of terms through len(β):

len(β) len(β) len(α) len(β) X X X X βi = n = αi + αi > αi, i=1 i=1 i=len(β)+1 i=1 so α 6 β.

Proposition 2.19. The minimal elements of Block(n) are precisely those where α is minimum in Dis(n, len(α)), and β is maximum in Dis(n, len(β)), and len(α) = len(β) + 1.

Proof. Let [α|β] ∈ Block(n), with the partition α minimum in Dis(n, len(α)), and the partition β maximum in Dis(n, len(β)), and len(α) = len(β) + 1. Let [α|β] majorize [α0|β0] and consider the form of the latter: since α majorizes only itself and some elements of length greater than len(α) by Lemma 2.18, and β is majorized only by itself and some elements of length shorter than len(β), it follows that α0 = α or α0 has length longer than α, and that β0 = β or has length shorter than β. As an element of Block(n), [α0|β0] must have a length diﬀerential of no more than one between α0 and β0. The only possible case then is [α0|β0] = [α|β], and hence [α|β] is minimal in Block(n). Furthermore, no other blocks are minimal: If α is not minimum in Dis(n, len(α)), then any block [α|β] majorizes [γ|β], where γ is the minimum element of Dis(n, len(α)). Similarly, if β is not maximum in Dis(n, len(β)), then any block [α|β] majorizes [α|γ] where γ is the maximum element of Dis(n, len(β)). Lastly, let [α|β] ∈ Block(n), with α minimum in Dis(n, len(α)), β maximum in Dis(n, len(β)), and len(α) 6= len(β) + 1. Thus len(α) = len(β). Let γ be the maximum element of Dis(n, len(β) − 1). Therefore, γ1 = β1 + β2, and γi = βi+1 for all 2 ≤ i ≤ len(γ), and so γ β. Thus [α|β] [α|γ], so [α|β] is not minimal.

26 However, the condition for a block to be minimal is not preserved within similarity equivalence classes. When α is shifted, α1 changes, but αm does not change. For example, the block [321|51] is minimal in Block(6), but [521|71] is not minimal in Block(8). We conclude with a summary of the principal ideas and results of this chapter. We first constructed posets of individual partitions of integers into distinct parts, and of ordered pairs of these partitions under majorization and length conditions. Since each block [α|β] is connected to a split graph degree sequence, and one or more split graphs which realize that sequence, we gave characterizations in terms of α and β of threshold, balanced, Nordhaus- Gaddum, and hairy split graphs, and minimal elements of Block(n). We also established an equivalence relation on blocks where two blocks are similar if some pair of graphs realizing each block differs only by the number of leaves adjacent to a specified vertex of highest degree. Graphs in each equivalence class are also consistent with respect to being threshold, balanced, Nordhaus-Gaddum, or hairy. Several open questions remain, especially regarding the Hasse diagram visualizations of Block(n), the realization numbers of blocks, and the similarity relation, and are listed in Chapter 6.

27 Chapter 3

The Inverse Operation on Split Graphs

3.1 Introduction

In Chapter 2, we studied split graphs through a poset representation; we now shift perspective and study an inverse operation on split graphs. This operation is like the complement in that it is a type of involution that operates on split graphs, but the inverse acts on a particular clique-stable set partition of a graph. As a consequence, we will see that a split graph can have either a unique inverse or two distinct inverses.

Deﬁnition 3.1. The inverse of a split graph G with partition K ∪ S is the graph (K ∪ S)I with identical vertex set, and an edge set comprising no edges in K, all edges in S, and the same edges between K and S as in G. Let I(G) denote the set of all (one or two) inverses of G.

The inverse diﬀers from the complement in that the complement contains the edges between K and S not present in G, whereas the inverse maintains the same edges between K and S as the original graph. An example is given in Figure 3.1. In [17], Tyshkevich lists several properties about the inverse operation without proof; in this chapter, we provide proofs of these and other new properties, some of which are necessary for subsequent chapters. Before proceeding, we introduce shorthand notation for decompositions of graphs with

28 Figure 3.1: The complement G, inverse GI , and inverse complement GI of a split graph with a ﬁxed KS-partition dominating, swing, or isolated vertices. Any vertex of these three types becomes its own component in a graph decomposition. Since all indecomposable split graphs are balanced (and hence have a unique KS-partition), or single vertex graphs, we are able to fully specify a composition by writing a single vertex graph as K1 to notate that it is in the clique part of the composition, or S1 to notate that it is in the stable set. The partition assigned to swing vertices is immaterial, so we denote these as K1 consistently but arbitrarily. In all instances besides written-out compositions, all single vertex graphs are referred to as K1.

3.2 Inverses Depend on the KS-Partition

Since the inverse operates with respect to a particular clique-stable set partition of a split graph, it is not necessarily unique. However, Tyshkevich [17] has determined exactly when it will be unique, and we provide a proof for completeness.

I Proposition 3.1. The inverse (K ∪ S) is unique if and only if G is balanced or K1. If the inverse of G is unique, then we denote the inverse by GI .

Proof. The inverse operation is ﬁxed with respect to the KS-partition of G, and balanced graphs have only one such partition, up to isomorphism. All unbalanced graphs have two

29 non-isomorphic KS-partitions, a K-max partition K ∪ S and an S-max partition K0 ∪ S0. By Theorem 1.5, the two cliques have diﬀerent sizes, that is, |K| 6= |K0|. The ﬁrst inverse |K| |S| |K0| |S0| deletes 2 edges and adds 2 edges, and the second deletes 2 edges and adds 2 edges. Since |K|= 6 |K0|, there can only be equal numbers of deleted edges in the two inverses |K| |S| if 2 = 0 and 2 = 0. The two inverses are thus isomorphic exactly when this condition holds. This occurs only when all of K, K0, S, and S0 are less than 2, which is precisely the single vertex graph.

Figure 3.2 gives an example of a graph, P4 ◦ K1, with two non-isomorphic inverses. Two corollaries arise from the above proposition:

Figure 3.2: The bull graph, P4 ◦ K1, with two partitions, and its two inverses

Corollary 3.2. [17] Indecomposable split graphs have unique inverses.

Proof. All indecomposable split graphs are balanced or K1.

Corollary 3.3. All threshold graphs with 2 or more vertices have non-unique inverses.

Proof. Threshold graphs are unbalanced.

30 3.3 Involutions

Tyshkevich [17] states and we conﬁrm that the inverse operation, applied twice, returns the original graph.

Proposition 3.4. Each inverse is an involution, i.e., ((K ∪ S)I )I = K ∪ S.

Proof. Consider G with a ﬁxed KS-partition. Taking the inverse twice with respect to this partition will yield ((K ∪ S)I )I = K ∪ S again, since the clique and stable sets are switched and return to the original, but no changes occur between K and S. Switching the partition for the second inverse will ensure ((K ∪ S)I )I 6= K ∪ S, since the clique and stable set sizes will change, as shown in the proof of Proposition 3.1.

Hence if a graph has a unique inverse, this inverse will act as an involution. When a graph has two possible inverses, they are separately, but not jointly, involutions. This is unlike the complement involution, which is unique to a graph. When composed, the inverse and complement operations form the inverse complement, and the order of composition is immaterial.

I Proposition 3.5. The inverse and complement operations commute: (K ∪ S)I = (K ∪ S) .

I Proof. Both (K ∪ S)I and (K ∪ S) have edge sets consisting of all edges in K, no edges in S, and the complement of the edges between K and S. Hence the two are isomorphic.

We can calculate how each of these involutions, including the inverse complement, change the degree sequence of G.

Proposition 3.6. If G is balanced with degree sequence d(G) = d1, d2, . . . , dh, dh+1, . . . , dn, where h is the clique number of G, then the degree sequences of G, GI , and GI are:

d(G) = n − 1 − dn, . . . , n − 1 − d1,

I d(G ) = dh+1 + n − (h + 1), . . . , dn + n − (h + 1), d1 − (h + 1), . . . , dh − (h + 1),

I and d(G ) = n + h − dh, . . . , n + h − d1, h − dn, . . . , h − dh+1.

31 3.4 Inverses of Tyshkevich Compositions

Understanding the relationship between Tyshkevich decomposition and the inverse operation is important for results in this chapter and subsequent chapters. In [17], Tyshkevich claims that the inverse of the composition of two graphs reverses their order. We provide a proof for completeness.

Theorem 3.7. For any split graphs G, H, the inverse with respect to composition follows the shoes-and-socks property: ((K ∪ S) ◦ (K0 ∪ S0))I = (K0 ∪ S0)I ◦ (K ∪ S)I . For balanced graphs, we can thus write (G ◦ H)I = HI ◦ GI .

Proof. Let G and H be split graphs, and ﬁx G with KS-partition K ∪ S, and H with KS-partition K0 ∪ S0. Then, (K ∪ S) ◦ (K0 ∪ S0) has edge set {xy} where

• x, y ∈ K or K0,

• x ∈ K, y ∈ S0,

• x ∈ K, y ∈ S, and xy ∈ E(G), or

• x ∈ K0, y ∈ S0, and xy ∈ E(H).

Thus, (G ◦ H)I has edge set {xy} where

• x, y ∈ S or S0,

• x ∈ S, y ∈ K0,

• x ∈ S, y ∈ K, and xy ∈ E(G), or

• x ∈ S0, y ∈ K0, and xy ∈ E(H), which is identical to the edge set of (K0 ∪ S0)I ◦ (K ∪ S)I . This is represented in Figure 3.3. Therefore, ((K ∪ S) ◦ (K0 ∪ S0))I = (K0 ∪ S0)I ◦ (K ∪ S)I .

By induction, we can extend Theorem 3.7 to a theorem for the inverse of the composition of n + 1 split graphs.

Theorem 3.8. If G = Gn ◦ ... ◦ G0, then I(G) = {A0 ◦ ... ◦ An|Ai ∈ I(Gi)}.

32 Figure 3.3: Block diagram of the interaction of inverses and the composition operation. The edge between two vertex sets is solid where all edges between any vertices of those sets are present, dashed if some edges exist, and dotted where no edges are allowed.

Thus, the inverse of the Tyshkevich decomposition of G into indecomposable com-

I I ponents is I(G) = {A0 ◦ G1 ◦ ... ◦ Gn|A0 ∈ I(G0)}. We can use this theorem to determine whether a graph composition has a unique inverse, and if the inverses of a graph composition are balanced.

Proposition 3.9. The set I(G ◦ H) has one element if and only if H is balanced and GI is

unique (that is, when G is balanced, or K1).

Proof. We consider several cases. If either G or H lacks a unique inverse, then so too does G ◦ H. If both G and H are balanced, then each element of their decomposition has a ﬁxed KS-partition (there are no swing vertices), so the inverse of G ◦ H is unique. If H is a single

I vertex and G is unique, then H has both K1 and S1 as inverses, where K1 and S1 are the I I single vertex graph with diﬀerent split partitions. Since K1 ◦G 6= S1 ◦G , the inverse of G◦H is not unique. Lastly, if G is a single vertex and H is balanced, then (G ◦ H)I = HI ◦ GI is unique, since GI is a swing vertex, and the decomposition is ﬁxed regardless of the partition of GI .

Proposition 3.10. Every graph of I(G) is unbalanced if and only if G has a dominating or isolated vertex; otherwise, they are all balanced.

33 I I Proof. Since G = K1 ◦ H or G = S1 ◦ H, any inverse of G takes the form H ◦ K1, where H is some inverse of H. Thus, each inverse has a swing vertex and is unbalanced.

3.5 Properties Preserved by the Inverse

In this section, we consider whether the inverse operation preserves various properties of split graphs. Many properties are not preserved; for instance, the inverse of a balanced graph

may be unbalanced. The bull graph (P4 ◦ K1) in Figure 3.2, which is unbalanced, is a good

example. By Theorem 3.8, its inverses are S1 ◦ P4 and K1 ◦ P4, both of which are balanced. Similarly, none of the Nordhaus-Gaddum types of split graphs (neither type NG-1, nor type NG-2, nor their intersection) are preserved under inverses. There are unbalanced graphs in all three types that have a balanced inverse, and balanced graphs are never Nordhaus-

Gaddum. Again, the bull graph (P4 ◦ K1), which is NG-1 and NG-2, is a good example of this. However, both indecomposability and being a unigraph are invariant under inverses. The former is stated in [17], and we prove both claims here.

Proposition 3.11. A split graph G is indecomposable if and only if GI is indecomposable.

Proof. Let G be indecomposable. If, for a contradiction, GI is decomposable with GI = (K ∪S)◦(K0 ∪S0), then by Theorem 3.8, (GI )I = (K0 ∪S0)I ◦(K ∪S)I . Since (GI )I = G, we have just decomposed G. Therefore, GI is indecomposable. The reverse direction is similar. An alternate proof uses Theorem 1.8 and a proposition from later in this chapter.

I Since G is indecomposable, there exists a P4-covering of G that is connected. Since P4 = P4 , I I the inverse of each P4 covers the same 4 vertices of G by Lemma 3.14. Therefore, G has a I I connected P4-covering, and is indecomposable. By Proposition 3.4, (G ) = G, so the above proof holds to show that if GI is indecomposable, so too is G. We can also use this argument to show that indecomposability is invariant under complementation, regardless of if G is

split, since 2K2 and C4 are each other’s complements, and P4 is self-complementary.

Proposition 3.12. A split graph G is unigraphic if and only if its inverse(s) is unigraphic.

34 Proof. First, let G be indecomposable, so its inverse is unique. If G is not unigraphic, then let H be a non-isomorphic graph realizing the same degree sequence. Since there exists a degree sequence characterization of indecomposable graphs, given in [17], it follows that H must also be indecomposable. Hence H also has a unique inverse. By Proposition 3.6, the degree sequence of the inverses of graphs realizing balanced degree sequence is ﬁxed, so GI and HI realize the same degree sequence. If, for a contradiction, GI ∼= HI , then it follows that (GI )I ∼= (HI )I , that is, G ∼= H. Hence, if G and H are not isomorphic, then neither are their inverses. The reverse direction follows easily: if G ∼= H, then clearly GI ∼= HI . Therefore an indecomposable graph is unigraphic if and only if its inverse is as well.

Let G be any split graph, with decomposition G = Gn ◦ ... ◦ G0. If G is unigraphic, then by Theorem 1.12, so too is every Gi. Since Gi is indecomposable, by the above argument I I I I Gi is also unigraphic. Thus any inverse of G is the composition G0 ◦ ... ◦ Gn, where G0 may be balanced (and unique), or one of K1 or S1. Since these components are each unigraphic, by Theorem 1.12, their composition, an inverse of G, is unigraphic. Similarly, if (K ∪ S)I is unigraphic, then we apply the above argument to conclude that ((K ∪ S)I )I = G is unigraphic.

The result is slightly stronger: if any inverse of G is unigraphic, all its inverses are.

3.6 Hereditary Classes and the Inverse

Hereditary subclasses of split graphs are not necessarily closed under the inverse operation. For instance, the class of hairy graphs, which is hereditary, is not closed; a good counterex-

ample is the bull graph in Figure 3.2. This graph is hairy, but one of its inverses, K1 ◦ P4, is not. Nevertheless, under certain conditions regarding the forbidden induced subgraphs of a hereditary class, we can show that a hereditary class will be closed under inverses. We begin with two lemmas about induced subgraphs that will be necessary for this work; ﬁrstly, that balanced induced subgraphs inherit the KS-partition of the main split graph, and secondly, that balanced induced subgraphs commute with inverses.

Lemma 3.13. Let G be a split graph and let H be a balanced induced subgraph of G. Let

35 K0 ∪ S0 be the unique KS-partition of H. For any KS-partition of G, it holds that K0 ⊆ K and S0 ⊆ S.

0 Proof. Let k1 ∈ K . Since H is balanced, k1 is not a swing vertex, so there exists a vertex 0 0 s ∈ S with k1s ∈ E(H). There also exists a vertex k2 ∈ K with k2s 6∈ E(H) since s is not

a swing vertex. This conﬁguration of three vertices and their edges is induced in G; since k1 0 has two non-adjacent neighbors in G, it follows that k1 ∈ K. Thus K ⊆ K. The proof that S0 ⊆ S is similar.

Lemma 3.14. If H is induced in G and H is balanced, then HI is the induced subgraph of every graph of I(G) on the vertex set V (H).

Proof. Let H be a balanced induced subgraph of G, and let (K ∪ S)I be an inverse of G. Let H have split partition K0 ∪ S0 and let x, y ∈ V (H). If x and y are both clique vertices in H, then xy ∈ E(H) and xy 6∈ E(HI ). By Lemma 3.13, x, y ∈ K, so xy is not an edge of (K ∪ S)I . Similarly, if x, y ∈ S0 and so xy ∈ E(HI ), then by Lemma 3.13, S0 ⊆ S, so x, y ∈ S. Thus xy ∈ E((K ∪ S)I ). Lastly, if x ∈ K0 and y ∈ S0, then xy ∈ E(HI ) if and only if xy ∈ E(H). Since x ∈ K and y ∈ S, by Lemma 3.13, and H is induced in G, we conclude that xy ∈ E(G) and so xy ∈ E((K ∪ S)I ). Therefore the edges of E(HI ) are exactly the edges of E((K ∪ S)I ) on V (H), and this holds for all inverses of G.

For instance, in Figure 3.2, P4 is induced in the bull graph and each of its inverses. More generally, inverses and induced subgraphs commute with respect to partition.

Remark 3.15. Let H be an unbalanced induced subgraph of G. Any inverse (K ∪ S)I of G contains as an induced subgraph the inverse of H taken with respect to the partition K ∪ S.

From here, we show that the threshold graphs are closed under inverses, which will serve as an example of a generalized result.

Proposition 3.16. The class of threshold graphs is closed under inverses.

Proof. Recall from Theorem 1.6 that a graph G is threshold if and only if it contains no

induced P4, and also note that P4 is balanced. We will show the contrapositive of each direction. Firstly, we show that if G is not threshold, then all inverses of G are not threshold.

36 I By Lemma 3.14, if P4 is induced in G, then P4 = P4 is induced in all graphs of I(G). Thus, these inverse graphs are not threshold. Secondly, we show that if any inverse of G is not

I threshold, then G itself is not threshold. Let (K ∪ S) be an inverse of G; if P4 is induced I I I I in (K ∪ S) , then P4 = P4 is induced in all ((K ∪ S) ) . Taken with the same partition, ((K ∪ S)I )I = G, as shown in Proposition 3.4. Thus, G is threshold if and only if all its inverses are threshold, so the class of threshold graphs is closed under inverses.

I The graphs P4 and P4 are shown in Figure 3.4.

Figure 3.4: P4 is its own inverse

We present a generalization of Proposition 3.16 to any hereditary class whose forbidden induced subgraphs obey several criteria. The proof is akin to that of Proposition 3.16.

Theorem 3.17. If graphs H1,...,Hn are balanced split graphs with balanced inverses, and form a class closed under taking inverses, then the hereditary class of split graphs with no

induced H1,...,Hn is closed under inverses.

One corollary of this theorem is that the subclass of split graphs in the hereditary closure of unigraphs is closed under inverses. Proving this requires additional insights, and we will defer the proof to Section 5.3.

37 Chapter 4

Non-Unigraphs Induced in Indecomposable Unigraphs

4.1 Introduction

The class of unigraphs is not hereditary, and Figure 4.5 shows an example of two non-

unigraphs induced in the graph S3(1, 2, 1), which is unigraphic. A natural question, then, is to investigate hereditary classes as close as possible to the class of unigraphs. In 2012 and 2013, Barrus [1], [2] gave structural, forbidden induced subgraph, and degree sequence characterizations for the largest hereditary class contained within the unigraphs. We consider the opposite direction: what is the smallest hereditary class containing all unigraphs? We give a structural characterization of this class – the hereditary closure of the unigraphs – in terms of graph decompositions. This characterization uses both the class of indecomposable unigraphs and the class of indecomposable non-unigraphs induced in indecomposable unigraphs. Tyshkevich [17] has identified the former class. In preparation for the statement and proof of the characterization in Chapter 5, we use this chapter to determine all graphs in the latter class. Tyshkevich classified all indecomposable unigraphs with nine graphs and families, and we will utilize this characterization, family by family, to identify the non-unigraphs induced in each. Her classification also includes the complement and, for split graphs, the inverse and inverse complement of each graph. Recall Definition 3.1 and Figure 3.1 for the definition

38 of the inverse operation. We already have all the tools in place to show that if G is an indecomposable unigraph, so too is its complement, and any inverses and inverse complements that exist. In the case of the inverse and inverse complement, these follow from Propositions 3.11 and 3.12. Furthermore, this applies to indecomposable non-unigraphs induced in indecomposable unigraphs. Proposition 4.1 is a corollary of the commutativity of induced subgraphs with the complement and inverse operations. The latter is shown in Lemma 3.14, which requires these induced non-unigraphs to be indecomposable.

Proposition 4.1. If G is an indecomposable non-unigraph induced in some indecomposable unigraph, then so too is G, as well as GI and GI when G is split.

Moreover, the complete list of induced indecomposable non-unigraphs can be derived by taking complements, inverses, and inverse complements (as applicable) of the indecomposable non-unigraphs induced in the list of nine indecomposable unigraphs and graph families. We introduce a notational convenience. In a split graph G with a ﬁxed KS-partition, for any vertex k ∈ K, let S-degG(k) denote the number of vertices in S adjacent to k, and observe that S-degG(k) = deg(k) − |K| + 1. Unless otherwise speciﬁed, we consider induced subgraphs to have the inherited KS-partition.

4.2 Non-Split Graphs

In this section, we consider each of Tyshkevich’s classes of indecomposable non-split unigraphs and ﬁnd all of the non-unigraphs induced in them.

4.2.1 The cycle C5

Figure 4.1 shows the graph C5. The induced subgraphs of C5 are K1, K2, P3, P4, 2K1, and

K2 ∪ K1, where ∪ represents the disjoint union. All are unigraphs.

39 Figure 4.1: The ﬁrst three types of indecomposable non-split unigraphs

4.2.2 The family mK2

Figure 4.1 shows the graph family mK2, which is the disjoint union of m copies of K2. Every

induced subgraph is thus of the form aK1 ∪ bK2 for some natural numbers a, b, where we

write aK1 as shorthand for a copies of K1, composed: K1 ◦ K1 ◦ ... ◦ K1. Since every vertex

of aK1 ∪ bK2 is of degree 0 or 1, it is unigraphic.

4.2.3 The family U2(m, n)

Figure 4.1 shows the graph U2(m, n), which is the disjoint union of mK2 and K(n, 1), the n-star. Any induced subgraph of U2(m, n) takes the form aK1 ∪ bK2 ∪ K(c, 1). Since only one vertex has degree ≥ 2, it follows that it is unigraphic.

4.2.4 The family U3(m)

Figure 4.2 shows the graph family U3(m), which is formed by attaching one C4 and m copies of C3 at a shared vertex. For all pairs (s, t) of natural numbers with s ≥ 0 and t ≥ 0, the graphs A(s, t) and B(s, t) in Figure 4.2 are non-isomorphic realizations of the degree sequence (2s + t + 3), 2,..., 2, 1,..., 1. Both are induced subgraphs of U3(s + t + 1), taking | {z } | {z } 2s+3 t+1 s + 1 or s copies of C3, t or t + 1 individual vertices from copies of C3, and 3 or 4 vertices

from the C4 subgraph, respectively.

40 Figure 4.2: The class of indecomposable unigraphs U3(m) and its two induced subgraph classes that are not unigraphs, A(s, t) and B(s, t).

Proposition 4.2. The only induced subgraphs of U3(m) that are not unigraphic are the elements of graph families A(s, t) and B(s, t) for s, t ≥ 0.

Proof. Let H be an induced subgraph of U3(m). Consider vertices a, b, c, d ∈ U3(m) as labeled in Figure 4.2. If c 6∈ V (H), then there is at most one vertex of degree 2 in V (H), namely a. Thus H is an induced subgraph of U2(m, n) and so is unigraphic. Otherwise, let c ∈ V (H). If a 6∈ V (H), then H = K1 ◦ xS1 ◦ yK2, with x degree 1 vertices and y triangles adjacent to c. Since all the components of this composition are unigraphic, so too is H, by Tyshkevich’s unigraph theorem. Otherwise, let a ∈ V (H). If a is an isolated vertex, then the above argument applies, appending S1 at the front of the composition. Thus, up to symmetry of b and d, let b ∈ V (H), so that a is not disconnected from the graph. There are several possibilities:

• If H contains neither C4 nor any C3, then H = S2(degH (c) − 1, 1, 1, 1), the split graph with two clique vertices, one of which has one attached leaf, and the other

has degH (c) − 1 leaves attached. Tyshkevich has shown that this is unigraphic and indecomposable (see subsection 4.3.3.)

• If H contains some C3, but not C4, then H = A(s, t) for some s, t ≥ 0.

41 • If H contains C4, but c has no leaf vertices, then either H = C4 or H = U3(k) for some k, and is unigraphic in either case.

• If H contains C4 and c has at least one leaf, then H = B(s, t) for some s, t ≥ 0.

Furthermore, every non-unigraph A(s, t) or B(s, t) induced in U3(m) for some m is indecomposable. We prove this using Theorem 1.8 and show that both non-unigraphs have

connected A4 graphs.

Proposition 4.3. Both A(s, t) and B(s, t) are indecomposable for all choices of s ≥ 0 and t ≥ 0.

Proof. We show that all vertices in either graph are induced in a P4 or C4 shared with vertex c. Let v be any vertex other than a, b, c, or d, so v is either part of one of the triangles, or a

leaf vertex of c. Then, {a, b, c, v} is an induced P4 including vertex c. This accounts for all

vertices but d in B(s, t), which is part of the induced C4 including vertex c. Therefore, both

A(s, t) and B(s, t) have connected A4 graphs, so are indecomposable.

4.3 Split Graphs

We now address the graph and four families of indecomposable unigraphic split graphs. We adopt Tyshkevich’s notation of these graph families, but note that the subscripts in S(p, q),

S2(p1, q1, . . . , pr, qr), S3(p, q1, q2), and S4(p, q) are not parameters, only means to keep track of each class. Before proceeding, we introduce shorthand notation for decompositions of graphs with multiple dominating, swing, or isolated vertices. If G = Gn ◦ ... ◦ G0 and G has c dominating vertices, then Gn = ... = Gn−c+1 = K1, because each of the dominating vertices is in the clique part of the decomposition partition. In this instance, we use shorthand to write G’s decomposition G = cK1 ◦ Gn−c ◦ ... ◦ G0, where cK1 = K1 ◦ ... ◦ K1, for c copies of K1. Similarly, if G has c isolated vertices, then Gn = ... = Gn−c+1 = S1, and we write

G = cS1 ◦ Gn−c ◦ ... ◦ G0, where cS1 = S1 ◦ ... ◦ S1, for c copies of S1. Lastly, if G has c

42 swing vertices, then Gc = ... = G0 = K1, and we write G = Gn−c ... ◦ ...G1 ◦ cK1 (the partition assigned to swing vertices is immaterial, so we denote these as K1 consistently but arbitrarily.) Ultimately, we care most about indecomposable graphs, so this shorthand allows us to maintain focus on nontrivial components of a decomposition.

4.3.1 The graph K1

The single vertex graph has only itself and the empty graph as induced subgraphs.

4.3.2 The family S(p, q)

The graph S(p, q) with p ≥ 1 and q ≥ 1, shown in Figure 4.3, is the split graph with q clique vertices, each of which is adjacent to exactly p leaves. This is precisely the graph S2(p1, q1) with r = 1, also shown in Figure 4.3. In Section 4.3.3, we extend the class S2(p1, q1, . . . , pr, qr) to include r = 1, and prove that all induced subgraphs of S(p, q) and S2(p1, q1, . . . , pr, qr) are unigraphic.

Figure 4.3: Two types of indecomposable split unigraphs. The circled vertices form the graphs’ cliques, although clique edges are not shown. The lower row of vertices in each graph are the stable set vertices.

43 4.3.3 The family S2(p1, q1, . . . , pr, qr)

Figure 4.3 shows the graph family S2(p1, q1, . . . , pr, qr) with pi ≥ 1, qi ≥ 1, and r ≥ 2. This is the split graph with q1 + q2 + ... + qr clique vertices, and for all i ≤ r, those qi clique vertices are adjacent to exactly pi leaves. The pi need not be distinct. For instance, S2(2, 3, 2, 3) and S2(2, 6) both denote the graph with six clique vertices, each of whom is adjacent to two leaves. We extend this class to include r = 1, as described in Section 4.3.2.

Proposition 4.4. Every induced subgraph of S2(p1, q1, . . . , pr, qr) is unigraphic.

Proof. Let H be an induced subgraph of S2(p1, q1, . . . , pr, qr), with KS-partition K∪S. Since split graphs are a hereditary class, H is a split graph. While S2(p1, q1, . . . , pr, qr) has no swing vertices or isolated vertices, it is possible that H does. However, the induced subgraph of

H with all swing and isolated vertices removed is in the form of an S2(p1, q1, . . . , pr, qr) graph, so we can decompose H into this induced subgraph, along with swing and isolated vertices. Consider H with the inherited partition from S2(p1, q1, . . . , pr, qr), and let W be the set of swing vertices in K. Order the vertices in K − W as k1, k2, . . . , k|K−W |. Then

H = nS1 ◦ S2(S-degH (k1), 1,...,S-degH (k|K−W |), 1) ◦ |W |K1 for n many isolated vertices; by the deﬁnition of S2(p1, q1, . . . , pr, qr), the pi values need not be distinct. If K − W is empty, then all vertices are swing or isolated, so H = nS1 ◦ |W |K1. In both cases, all components of the decomposition are unigraphic, so H is a unigraph.

In essence, S2(p1, q1, . . . , pr, qr) is almost a hereditary class, save for swing vertices and isolated vertices. An alternate proof comes from constructing H from its degree sequence: |K| is ﬁxed in a K-max partition, and the number of leaves per vertex is also ﬁxed, completing the unique construction.

4.3.4 The family S3(p, q1, q2)

Figure 4.4 shows the graph family S3(p, q1, q2), with p ≥ 1, q1 ≥ 2 and q2 ≥ 1. These graphs are split graphs with a clique Q containing q1 + q2 vertices. Let Q = Q1 ∪ Q2, with |Q1| = q1 and |Q2| = q2. All vertices in Q have degree p+1, but those in Q2 are adjacent to p+1 leaves,

44 while those in Q1 are adjacent to p leaves and e. The smallest example of graphs in this family is S3(1, 2, 1), which is depicted in Figure 4.5, along with its induced non-unigraphs.

Figure 4.4: The third type of indecomposable split unigraphs. The largest circle of vertices forms the graph’s maximum clique, although clique edges are not shown.

The induced subgraphs of S3(p, q1, q2), save for isolated vertices, look like elements of S2(p1, q1, . . . , pr, qr) or S3(p, q1, q2) with variable numbers of leaf vertices. Those that are subgraphs of S2(p1, q1, . . . , pr, qr) are unigraphic as above. Of those containing e, if two clique vertices, one adjacent to e and one not, have diﬀerent degrees, then it is possible to shift leaves from one to the other to create a non-isomorphic graph with the same degree sequence. We formalize this idea in Proposition 4.5 and illustrate it in Figure 4.5.

Proposition 4.5. Let H be an induced subgraph of S3(p, q1, q2), with KS-partition K ∪ S. The graph H is a non-unigraph if and only if three conditions are met:

1. e ∈ V (H)

2. degH (e) ≥ 2

3. There exist clique vertices x, y ∈ K where x is adjacent to e, y is not adjacent to e, y

is adjacent to at least one leaf, and degH (x) 6= degH (y).

Proof. If H meets the conditions, then let S-degH (x) = m and S-degH (y) = n. Thus, since x is adjacent to e, x has m − 1 leaves and y has n leaves. We know that m ≥ 1 and n ≥ 1 since

45 Figure 4.5: S3(1, 2, 1), the smallest graph in S3(p, q1, q2), and two non-unigraphs induced in

S3(1, 2, 1). we have assumed that y has at least one leaf. Create a graph H0 with V (H0) = V (H), and E(H0) = E(H), but remove n−m leaf edges from y and create edges from these n−m degree

0 1 vertices to x. If n − m is negative, then the leaves shift from x to y. In H , S-degH0 (x) = n and S-degH0 (y) = m, and the degree of all other vertices remains constant. Hence the degree sequence of H0 equals that of H. However, H0 and H are not isomorphic. In both graphs, e is the unique vertex of degree ≥ 2 not adjacent to every clique vertex but, potentially, itself. Note that y is a clique vertex in all partitions since S-degH (y) = n ≥ 1 and S-degH0 (y) = m ≥ 1, so y is never a swing vertex. Hence any isomorphism must map e to itself. Nevertheless, the degree

0 sequence of NH (e) does not equal that of NH0 (e), so H and H are not isomorphic. Hence, H 0 (and H ) are non-unigraphic induced subgraphs of S3(p, q1, q2) for some choice of parameters. If H fails to meet the conditions, then we consider several possibilities, by condition.

If e 6∈ V (H), then H is induced in S(p, |K|). Otherwise, let e ∈ V (H), and let n = degH (e). If n = 1, then e is a leaf vertex, and H is again induced in S(p, |K|). If n = |K|, then K ∪e is a clique, and H is induced in S(p, |K| + 1). All induced subgraphs of S(p, q) are unigraphic, as shown in Section 4.3.2, so H is unigraphic.

Otherwise, 2 ≤ n < |K|, so there exist clique vertices x1, . . . xn (minimum 2) adjacent

46 to e, and y1, . . . y|K|−n (minimum 1) not adjacent to e. If every yj is adjacent to zero leaves, then e, y1, . . . , y|K|−n are all swing vertices, and H is induced in S(p, |K| + 1), and is unigraphic. Otherwise, there exists some yj with at least one leaf. The only remaining condition is that some xi and yj are of diﬀerent degree; to fail this condition, it follows that S-degH xi = S-degH yj for all xi adjacent to e and for all yj that are not swing vertices.

Therefore, H is nearly an element of S3(p, q1, q2), for some choice of parameters, save for swing vertices and isolated vertices. We can thus decompose H into aS1 ◦ S3(S-degH xi − 1, n, m) ◦

(|K| − n − m)K1, for a isolated vertices and |K| − n − m swing vertices. All the elements of the composition are unigraphic, so we conclude that H is unigraphic.

From here, we consider the subset of non-unigraphs induced in S3(p, q1, q2) that are indecomposable. We ﬁrst eliminate obvious issues, then apply Theorem 1.8.

Proposition 4.6. An induced non-unigraph of S3(p, q1, q2) is indecomposable if and only if it contains no isolated or swing vertices.

Proof. Since swing vertices are always decomposed as K1 at the far right of the decomposition, and isolated vertices are decomposed as S1 at the far left, any graph with swing and isolated vertices is decomposable.

For the other direction, let H be an induced non-unigraph of S3(p, q1, q2) containing neither isolated nor swing vertices. Since H is split, it has no induced C4 or 2K2 (see

Theorem 1.2), so we will use only induced P4 subgraphs to create A4(H). Since H has no swing vertices, it is balanced, so consider H with its single KS-partition inherited from

S3(p, q1, q2). Let x ∈ K be adjacent to e and let y ∈ K not be adjacent to e. Since y is not a swing vertex, it must be adjacent to some leaf s ∈ S. Then {e, x, y, s} induces P4 in H. This connects all clique vertices to e in the graph A4(H), as well as all stable set vertices adjacent to y. Lastly, let s0 ∈ S be a leaf adjacent to a clique vertex k with k 6= y. Then {s0, k, y, s} induces P4 in H, which connects all remaining leaf vertices to y in A4(H). Therefore A4(H) is connected and H is indecomposable by Theorem 1.8.

47 4.3.5 The family S4(p, q)

Figure 4.6 shows the graph family S4(p, q). It is formed from S3(p, 2, q), with p ≥ 1 and q ≥ 1, and a vertex f added to the clique, where f is adjacent to every stable set vertex but e. Let Q be the subset of clique vertices not adjacent to e.

Figure 4.6: The fourth type of indecomposable split unigraphs. The largest circle of vertices forms the graph’s maximum clique, although clique edges are not shown.

Proposition 4.7. Let H be an induced subgraph of S4(p, q), with K-max partition K ∪ S. The graph H is a non-unigraph if and only if at least one of the following conditions holds:

1. H is a non-unigraph induced in S3(p, 2, q).

2. H is not induced in any S3 graph; N(e) = {x1, x2}; and there exists a clique vertex

qi ∈ Q, with S-degH qi ≥ 1 and either S-degH qi 6= S-degH x1, or S-degH qi 6= S-degH x2.

3. H is not induced in any S3 graph; H contains a leaf vertex; x1 is adjacent to e; and

some vertex of degree 2 has a neighbor other than x1 and f.

Proof. If condition 1 holds, then H is obviously non-unigraphic.

Otherwise, whenever H is not induced in S3, H must contain at least two stable set vertices of degree 2, and so f ∈ H.

48 If condition 2 holds, then without loss of generality let S-degH qi = n and S-degH x1 = m, with n 6= m. In H, then, there are exactly m − 1 degree two vertices adjacent to x1 and f. This graph family is depicted in Figure 4.7. We will shift several degree two vertices between x1 and qi, similar to the shift of degree one vertices in the proof for S3.

Since x1 is adjacent to e, there are exactly m − 1 degree two vertices adjacent to x1 and f; and qi has n such vertices. We know that m ≥ 1 and n ≥ 1 since we have assumed 0 0 0 that S-degH qi ≥ 1. Create a graph H with V (H ) = V (H), and initially E(H ) = E(H), but remove n − m edges between qi and its degree two vertices, and create edges from these n − m degree two vertices to x1. Do not change the edges from these degree two vertices to 0 f. If n − m is negative, then the edges shift from x1 to qi. Then in H , S-degH0 (x1) = n and

S-degH0 (qi) = m, and the degrees of all other vertices remain constant. Hence the degree sequence of H0 equals that of H. However, H0 and H are not isomorphic. Let H0 have KS-partition K0 ∪ S0. We consider two cases: ﬁrst, if in both graphs f is the only vertex adjacent to all but one of

0 the stable set vertices (i.e., S-degH (f) = |S| − 1 and S-degH0 (f) = |S | − 1), then f must be ﬁxed in any isomorphism. Since e is the only vertex not adjacent to f, it too is ﬁxed.

0 However, the degrees of x1 and x2, the neighbors of e, change between H and H , so the graphs are not isomorphic. Alternatively, if there is another vertex adjacent to all but one

0 stable set vertex, then this vertex must be x1 or qi, since x1f, qif ∈ E(H),E(H ). If qi is the vertex in question in one of the graphs (up to symmetry, H), then e is the only vertex not adjacent to qi in H, so S-degH (x1) = S-degH (x2) = 1, and all clique vertices besides qi and 0 f are adjacent to no stable set vertices. Thus in H , the two vertices with S-degree 1 are x2 and qi, but they have no neighbors in common, so the graphs are not isomorphic. Lastly, if x1 is the vertex adjacent to all but one of the stable set vertices in one of the graphs, then qi is the vertex in question in the other graph. By the above logic, they are not isomorphic.

If condition 3 holds, then either degH (e) = 1 or f is adjacent to a degree one vertex.

First, we consider if degH (e) = 1, so N(e) = {x1}. Since H is not induced in S3, f must be adjacent to at least two degree two vertices, and at least one of these is not adjacent to x1. Let v, w ∈ S be degree two vertices with N(v) = {f, q} for q ∈ K and q 6= x1, and N(w) = {f, r}, with r ∈ K. Create a graph H0 from H by deleting edge rw and applying

49 Figure 4.7: The family of non-unigraphs induced in S4(p, q) failing condition 2, up to sym-

metry of x1 and x2, where m 6= n

one of the following changes:

(a) If r 6= x1, add edge er, making e into a degree two vertex and w into a degree one vertex. Now, the degree sequences of H and H0 are identical, so in a K-max partition, their cliques and stable sets are the same size. However, H and H0 are not isomorphic since

H includes a vertex, that is, x1, which has S-degH ≤ |S| − 2 and is adjacent to a leaf. On 0 the other hand, in H , all leaf vertices are adjacent to a vertex of S-degH0 = |S| − 1. 0 (b) If r = x1, add edge wq. Again, H and H have the same degree sequence, but are not isomorphic since S-degH0 x1 = S-degH x1 − 1. This is because x1 must be ﬁxed in any isomorphism: no vertices other than x1 and f can have leaves, so x1 must be mapped 0 to itself or f. However, x1 cannot be mapped to f since S-degH0 (f) = |S| − 1, and in H , x1 is adjacent to neither v nor w. Therefore, x1 is not mapped to f, and is instead ﬁxed; but this is not an isomorphism, since degH (x1) 6= degH0 (x1).

Second, if degH (e) = 2 and N(e) = {x1, x2}, then by condition 3, f is adjacent to at least one leaf, which we will call w, and f is adjacent to a degree two vertex, v, whose neighbor set is N(v) = {f, q}, for some q ∈ K.

(c) If q = x1, then the induced subgraph of H on vertices {x1, x2, f, e, v, w} is isomorphic to case (b) above. Create H0 by applying the switch described in case (b) to the

50 aforementioned vertex subset; since case (b) preserves the degree sequence but creates a non-isomorphic graph, so too H and H0 are non-isomorphic realizations of the same degree sequence.

0 (d) If q 6= x1, then create H by deleting edge ex2 and adding edge wx2. The graphs H and H0 have the same degree sequence. However, they are not isomorphic since in H0, f is adjacent to every degree two vertex, but in H, no vertex is adjacent to every degree two vertex (see that e and v have no neighbors in common).

On the other hand, if all the conditions fail, we will show that H must be unigraphic.

If H is induced in S3 and is unigraphic, then all conditions fail. Hence, we assume that H is not induced in S3; therefore, H must contain at least two stable set vertices of degree 2, and so f ∈ H. We consider cases by degH (e): If e 6∈ V (H), then H = K1 ◦ B, where K1 is the single vertex f, and B is some graph induced in S2. Since both components of the decomposition are unigraphs, so too is H.

If degH (e) = 1, then in order to contradict condition 3, all degree two vertices have neighbor set {x1, f}. Since K = {x1, f} ∪ T , where T is a set of swing vertices, it holds that H is unigraphic.

If degH (e) = 2, then since condition two fails, either H = S4(p, q) for some choice of p, q, or there exists no qi ∈ K not adjacent to e, with S-degH (qi) ≥ 1. Since the split

partition of H is K-max, K contains only {x1, x2, f}, and some number of swing vertices. If H contains a degree one vertex, then condition (c) applies above, and so condition 3 holds.

Thus all vertices in S have degree two. The degrees of {x1, x2, f} are suﬃcient to determine the placement of the stable set vertices. Speciﬁcally, the number of vertices adjacent to

exactly two of {x1, x2, f} is |S| − SdegH (k), where k is the third of those vertices. Thus, H is unigraphic.

From here, we restrict this list to only indecomposable non-unigraphs induced in

S4(p, q). Again, we are able to exclude any graph with isolated or swing vertices.

Proposition 4.8. An induced non-unigraph of S4(p, q) is indecomposable if and only if it contains no swing vertices.

51 Proof. As in the proof of Proposition 4.6, any graph with a swing vertex is decomposable.

Here, unlike for S3(p, q1, q2), no induced non-unigraph has isolated vertices, since every vertex besides e is adjacent to f, and e and f are included in every induced non-unigraph. For the other direction, let H be an induced non-unigraph, by Proposition 4.7, containing no swing vertices. Since H has no swing vertices, it is balanced, so consider H with its single KS-partition inherited from S4(p, q). If condition 1 from Proposition 4.7 holds, then H is induced in S3(p, 2, q), so by Proposition 4.6, H is indecomposable. Otherwise, if conditions 2 or 3 hold, we apply Theorem 1.8 and show that A4(H) is connected.

For both conditions 2 and 3, there must exist distinct vertices e, xi, f, k ∈ V (H)

where k ∈ K is not adjacent to e, and xi is adjacent to e. Since k is not a swing vertex, it is

adjacent to some s ∈ S, and by deﬁnition, N(s) = {k, f}. Thus, {e, xi, k, s} and {e, xi, f, s}

induce P4 in H. So far, we have connected e, all clique vertices, and some stable set vertices. We now account for any remaining stable set vertices. Let s0 ∈ S be adjacent to some

0 00 vertex k ∈ K not adjacent to e; then, {e, x1, k, s } forms an induced P4 in H. Let s ∈ S be 00 adjacent to f and a vertex xi adjacent to e; then, {e, xi, s , f} induces P4 in H. Lastly, if f

is adjacent to any leaf vertices l, then {e, xi, f, l} induces P4 in H.

Since every vertex in H is induced in a P4 subgraph including e, it follows that e is

adjacent to every vertex in A4(H). Therefore A4(H) is connected and, as a consequence of Theorem 1.8, H is indecomposable.

The graphs identiﬁed by Propositions 4.3, 4.6, and 4.8, taken together, are exactly the set of indecomposable non-unigraphs induced in an indecomposable unigraph.

52 Chapter 5

A Structural Characterization of the Hereditary Closure of the Unigraphs

5.1 Introduction

In this chapter, we give a characterization of the hereditary closure of the set of unigraphs. This class, the minimum hereditary class containing all unigraphs, is the class of all graphs induced in some unigraph. We begin by deﬁning two sets of graphs. Consider the set of unigraphs, and let U be the subset of indecomposable unigraphs. Tyshkevich [17] provides a complete list of the graphs in U, and our Theorem 1.12 presents her characterization. Thus, the set of unigraphs is the set of graphs that can be realized as a composition of graphs in U. Given the set of graphs induced in an element of U, let I be the subset of those graphs that are indecomposable. Hence an element of I is an indecomposable graph induced in a indecomposable unigraph. Since every graph is induced in itself, it follows that U is a subset of I. Furthermore, every unigraphic element of I is an element of U. Therefore, the elements of I are partitioned into two subclasses: the indecomposable unigraphs and the indecomposable non-unigraphs induced in indecomposable unigraphs. The former subclass is given by Tyshkevich, and the latter class is determined in Propositions 4.3, 4.6, and 4.8 of Chapter 4.

A note on vocabulary: the decomposition G = Gn ◦ ... ◦ G0 refers to the unique

53 Tyshkevich decomposition of a graph into indecomposable parts, whereas a composition

H = Hn ◦ ... ◦ H0 is deﬁned even when not all Hi are indecomposable.

5.2 Characterization Theorem

Before stating and proving the main theorem, we explain, state, and prove a lemma about Tyshkevich decomposition in general. Informally, if a graph H is induced in a graph G, then H is equal to the composition of its intersections with each component of the decomposition of G.

For example, let G = P4 ◦ C4:

Let H be the graph shown below, which is induced in G = P4 ◦ C4.

Then H = (P4 ∩ H) ◦ (C4 ∩ H).

54 We deﬁne notation for the induced intersection of two induced subgraphs. Let G be a graph with induced subgraphs A and B. Deﬁne the induced intersection of A and B, denoted A ∩ B, to be the graph induced in G by the vertex set V (A) ∩ V (B), and is induced in G. Since A and B are each induced in G, and V (A ∩ B) is a subset of both V (A) and V (B), it follows that A ∩ B is an induced subgraph of both A and B.

Lemma 5.1. Let G be a graph, with G = Gn◦...◦G0 and Gi has partition Ki∪Si when i ≥ 1.

Let H be an induced subgraph of G, and deﬁne Hi := H ∩ Gi to be the induced intersection 0 0 of H and Gi. If i ≥ 1, assign Hi the split partition Ki ∪ Si inherited from Gi. Then H is equal to the composition of its induced intersection components, that is, H = Hn ◦ ... ◦ H0.

Proof. We prove the lemma by induction on n. If n = 0, then G’s composition has only one component, in which case G0 = G. Hence H0 = H ∩ G = H. 0 Assume the lemma holds for n = k. Let n = k + 1, and consider G = Gk ◦ ... ◦ G0, ignoring the ﬁrst term of G’s composition. Let H0 := H ∩ G0 be the induced intersection

0 0 0 of H and G , and let Hi be the induced intersection of H and Gi. By the induction 0 0 0 hypothesis, it follows that H = Hk ◦ ... ◦ H0. We can substitute terms to return to Hi: 0 0 0 0 Hi = H ∩ Gi = H ∩ G ∩ Gi = H ∩ Gi = Hi. Therefore, it follows that H = Hk ◦ ... ◦ H0. 0 Since composition is associative, G = Gk+1 ◦ G , and it suffices to show that we can 0 append the remaining term Hk+1 to get H = Hk+1 ◦ H . 0 0 We first show the vertex sets of H and Hk+1 ◦H are equal: V (Hk+1 ◦H ) = V (Hk+1)∪ 0 0 0 V (H ) by definition of composition, and V (Hk+1) ∪ V (H ) = V (H ∩ Gk+1) ∪ V (H ∩ G ) = 0 0 V (H) ∩ V (Gk+1 ∪ G ) = V (H) ∩ V (Gk+1 ◦ G ) = V (H) ∩ V (G) = V (H). We now consider the edge sets:

0 Let edge xy ∈ E(H), for some x, y ∈ V (H) = V (Hk+1) ∪ V (H ).

• Case 1: x, y ∈ V (Hk+1).

0 Then xy ∈ E(Hk+1) ⊂ E(Hk+1 ◦ H ) since Hk+1 is induced in H.

• Case 2: x, y ∈ V (H0).

This is symmetric to case 1.

55 0 • Case 3: Up to symmetry, x ∈ V (Hk+1) and y ∈ V (H ).

0 Thus x ∈ V (Gk+1) and y ∈ V (G ). Since there are no edges between vertices of Sk+1 0 and V (G ), it follows that x ∈ Kk+1. Since induced split graphs inherit partitions, 0 0 x ∈ Kk+1 and xy ∈ E(Hk+1 ◦ H ).

0 For the reverse direction, let edge xy ∈ E(Hk+1 ◦ H ).

• Case 1: x, y ∈ V (Hk+1).

Then xy ∈ E(Hk+1). Since Hk+1 is induced in H, it follows that xy ∈ E(H).

• Case 2: x, y ∈ V (H0).

This is symmetric to case 1.

0 • Case 3: Up to symmetry, x ∈ V (Hk+1) and y ∈ V (H ).

0 0 By deﬁnition of composition, x ∈ Kk+1 ⊂ Kk+1. Since y ∈ V (G ), and all edges are 0 present in the composition between Kk+1 and V (G ), we conclude that xy ∈ E(Gk+1 ◦ G0) = E(G). Since H is induced in G, it follows that xy ∈ E(H).

0 Therefore E(H) = E(Hk+1 ◦ H ), and we conclude that the graphs are equal.

The components Gi need not individually be indecomposable, and regardless of

whether all Gi are indecomposable, the Hi components may or may not be indecomposable. Using this lemma, we give the characterization theorem.

Theorem 5.2. The hereditary closure of the set of unigraphs is exactly the set of graphs whose indecomposable components are in I, the set of indecomposable induced subgraphs of indecomposable unigraphs.

Proof. We begin by showing that if G is composed of elements of I, then G is an element of the hereditary closure of the set of unigraphs.

Let G = Gn ◦ ... ◦ G0 be the decomposition of G and let Gi ∈ I for all i ≤ n.

Since Gi ∈ I, there exists a unigraph Ui containing Gi as an induced subgraph. By the

deﬁnition of the composition operation, then, G = Gn ◦ ... ◦ G0 is induced in Un ◦ ... ◦ U0,

56 which is unigraphic by Theorem 1.12. Hence G is induced in a unigraph, and must be in the hereditary closure of the unigraphs. For the reverse direction, let G be induced in some unigraph U, which may not be

indecomposable. Decompose U into Un ◦ ... ◦ U0; by Theorem 1.12, each Ui is an inde-

composable unigraph. Let Gi = G ∩ Ui, so that each Gi is induced in Ui. By Lemma 5.1,

G = Gn ◦ ... ◦ G0.

Since each Gi is induced in Ui, we know that every Gi is an induced subgraph of an

indecomposable unigraph. However, each Gi might also not be indecomposable. For each i,

let Gi = Gi,mi ◦ ... ◦ Gi,0, where mi is the number of terms in the decomposition of Gi. Now,

for all j ≤ mi, Gi,j is induced in Gi, which in turn is induced in Ui. By transitivity of the

induced relation, it follows that Gi,j is induced in Ui. We conclude that Gi,j ∈ I for all i, j.

By substitution, the decomposition of G into indecomposable parts is G = (Gn,mn ◦

... ◦ Gn,0) ◦ ... ◦ (G0,m0 ◦ ... ◦ G0,0). Since the composition relation ◦ is associative, we

conclude that G = Gn,mn ◦ ... ◦ G0,0, and all the canonical components of G are in I.

5.3 Properties of the Hereditary Closure

We conclude this chapter with several observations and corollaries regarding Theorem 5.2. Since the hereditary closure of the unigraphs is hereditary, it will have a forbidden induced subgraph characterization. The elements of such a minimal class of forbidden induced subgraphs remain a primary open question of this work (see Chapter 6). However, we can draw some conclusions about these graphs, largely based on the above theorem and the minimality assumption.

Proposition 5.3. Let G be a minimal forbidden induced subgraph of the hereditary closure of the unigraphs. Then:

1. G is non-unigraphic.

2. G is indecomposable.

3. G and, if split, GI and GI , are also minimal forbidden induced subgraphs.

57 Proof. By deﬁnition, the class of unigraphs is contained in its hereditary closure, so any element outside the hereditary closure is non-unigraphic. If, for a contradiction, G is de-

composable, with decomposition G = Gn ◦ ... ◦ G0 where all Gi are indecomposable and non-empty, then all Gi are proper induced subgraphs of G. If any Gi is not an element of the hereditary closure of the unigraphs, then this contradicts the assumption that G is a minimal forbidden graph. Otherwise, every Gi is an element of the hereditary closure of the unigraphs, so by the ﬁrst part of the proof of Theorem 5.2, G is an element of the hereditary closure, and thus not forbidden. For the third claim, we ﬁrst show that G, and, when split, GI and GI are not elements of the hereditary closure, and subsequently that they are minimal forbidden induced subgraphs. We have shown in Proposition 1.1 and Lemma 3.14 that an indecomposable graph being induced in some other graph commutes with both the complement and the inverse op-

erations (for Lemma 3.14, we note that K1 is fortunately not a forbidden induced subgraph, so all indecomposable forbidden graphs are balanced). Additionally, both the complement and the inverse operations preserve the property of being a unigraph (see Proposition 3.12). Thus, it follows that G is induced in a unigraph exactly when any of G, and, when split, GI and GI are also induced in a unigraph. Since G is not an element of the hereditary closure, neither are G, nor, when split, GI nor GI . For minimality, we prove the contrapositive by cases. If G is not a minimal forbidden induced subgraph, then some minimal forbidden graph H is induced in G. We have shown in the above paragraph that hereditary closure is closed under complementation, so H is also a forbidden subgraph, and since complementation commutes with induced subgraphs, H is induced in G = G. Thus G is not minimal. If G is split and GI contains an induced minimal forbidden subgraph H, then by part 2 of this proposition, H is indecomposable. Therefore, HI is induced in (GI )I = G. Since the hereditary closure is also closed under inverses, HI is a forbidden subgraph, contradicting our assumption that G is minimal. The proof when GI is not minimal is a straightforward combination of the above two cases.

Since the class of graphs not induced in unigraphs are closed under complements and inverses, we state the same of the hereditary closure.

58 Corollary 5.4. The hereditary closure of the unigraphs is closed under complementation; and the subset of split graphs within this class is closed under inverses.

We hope these results will be useful for further work toward a forbidden induced subgraph characterization.

59 Chapter 6

Open Questions

There is ample opportunity for future research on the poset of blocks under majorization and its connection to split graphs. First, we would like to investigate the relationship between Tyshkevich decomposition and the similarity equivalence relation. A second question addresses the realization number r([α|β]) of a block: Is it possible to determine r([α|β]) from α and β? We hope to expand the preliminary work in 2.7 and precisely state the relationship between r([α|β]) and r([α0|β0]). Another enquiry relates to the Hasse diagram of Block(n). Is it possible to determine the row of [α|β] from α and β? It is known that the threshold graphs are the elements of the top row of Block(n), since all other graphs are majorized by them. Merris [14] has shown that the threshold covered graphs, that is, the second row of the Hasse diagram, are unigraphic. Could we at least characterize the covered graphs in terms of [α|β]? Another open question is whether the threshold graphs and covered graphs together form a hereditary class; if so, this idea could be extended downward to other rows of the Hasse diagram. It would be valuable to further investigate connections between the visual properties of the Hasse diagram and split graph properties of its elements. From Chapter 3, we are interested in ﬁnding out more about how the inverse acts on hereditary classes of graphs. Is the condition in Proposition 3.17 necessary, or are there hereditary classes closed under inverses whose forbidden induced subgraphs are either unbalanced or, together, not closed under inverses? Primarily, we are continuing to investigate other characterizations of the hereditary

60 closure of the unigraphs. Some preliminary considerations for a forbidden induced subgraph characterization are presented in Section 5.3, and we have also begun a computational search for forbidden induced subgraphs with small vertex sets. We also hope to ﬁnd a degree sequence characterization or show no such characterization is possible. Finally, we hope to analyze a point of clariﬁcation in Corollary 5.4. Is every split non-unigraph in the hereditary closure induced in a split unigraph? We could extend this line of inquiry about the hereditary closure of the split unigraphs to the consideration of various other subclasses of unigraphs.

61 References

[1] M. D. Barrus, On 2-switches and isomorphism classes, Discrete Math. 312 (2012), no. 15, 2217–2222.

[2] M. D. Barrus, Hereditary unigraphs and the Erd˝os-Gallai equalities, Discrete Math. 313 (2013), no. 21, 2469–2481.

[3] M. D. Barrus and D. B. West, The A4-structure of a graph, J. Graph Theory 71 (2012), no. 2, 159–175.

[4] V. Chv´ataland P. Hammer, Set-packing problems and threshold graphs, CORR 73-21, University of Waterloo, Canada, 1973.

[5] V. Chv´ataland P. Hammer, Aggregation of inequalities in integer programming, Ann. Discrete Math. 1 (1977), 145–162.

[6] C. Cheng, K. L. Collins and A. N. Trenk, Split graphs and Nordhaus–Gaddum graphs, Discrete Math. 339 (2016), 2345–2356.

[7] K. L. Collins and A.N. Trenk, Nordhaus-Gaddum Theorem for the Distinguishing Chromatic Number, Electron. J. of Combin. 20 (2013), no. 3.

[8] K. L. Collins and A. N. Trenk, Finding balance: split graphs and related classes, Electron. J. of Combin. 25 (2018), no. 1.

[9] K. L. Collins and A. N. Trenk, Split Graphs, Topics in Algorithmic Graph Theory, ed. L. W. Beineke, M. C. Golumbic, and R. J. Wilson, Cambridge University Press, Forthcoming.

62 [10] S. F¨oldesand P. Hammer, Split graphs, Congr. Numer. 19 (1977), 311–315.

[11] M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, 1980.

[12] P. Hammer and B. Simeone, The splittance of a graph, Combinatorica 1 (1981), 275–284.

[13] R. Merris, Graph Theory, Wiley-Interscience, 2001.

[14] R. Merris, Split graphs, European J. Combin. 24 (2003), 413–430.

[15] E. Ruch and I. Gutman, The branching extent of graphs, J. Combin. Inform. System Sci. 4 (1979), 285–295.

[16] R. Tyshkevich, The canonical decomposition of a graph, Dokl. Akad. Nauk. BSSR 24 (1980), 677–679 (in Russian).

[17] R. Tyshkevich, Decomposition of graphical sequences and unigraphs, Discrete Math. 220 (2000), 201–238.

[18] D. West, Introduction to Graph Theory, 2nd ed. Pearson, 2001.