    # Characterization by forbidden induced subgraphs of some subclasses of chordal graphs

Chordal graphs are the graphs in which every cycle of length at least four has a chord. A set S is a vertex separator for vertices a and b if the removal of S of the graph separates a and b into distinct connected components. A graph G is chordal if and only if every minimal vertex separator is a clique. We study subclasses of chordal graphs defined by restrictions imposed on the intersections of its minimal separator cliques. Our goal is to characterize them by forbidden induced subgraphs. Some of these classes have already been studied such as chordal graphs in which two minimal separators have no empty intersection if and only if they are equal. Those graphs are known as strictly chordal graphs and they were first introduced as block duplicate graphs by Golumbic and Peled. They were also considered in other previous works, showing that strictly chordal graphs are exactly the (gem, dart)-free graphs.

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## 1 Introduction

Throughout this text, a graph is always simple, finite and undirected. For sets and , we denote if is a subset of , and if is a proper subset of . For a graph we denote the set of vertices by and the set of edges by . Given two non-adjacent vertices and in the same connected component of , a uv-separator is a set contained in such that and are in different connected components of . This separator is minimal if no proper subset of is also a uv-separator. We will just say minimal vertex separator to refer to a set that is a -minimal separator for some pair of non-adjacent vertices and in . A graph is chordal if it has no cycle of length at least four as induced graph. A clique is a maximal set of pairwise adjacent vertices. A clique tree of a connected chordal graph is any tree whose vertices are the cliques of such that for every two cliques each clique on the path from to in contains .

In this paper we study the relationship between the family of minimal vertex separators and the structure of the graph. In particular we characterize the graph classes arising from properties imposed on the family of minimal vertex separators.

An important characterization of chordal graphs is due to Dirac: [] A graph is chordal if and only if every minimal vertex separator of is a clique. Two cliques of G form a separating pair if is non-empty, and every path in G from a vertex of to a vertex of contains a vertex of . Every minimal separator of a chordal graph G is a clique and moreover, it is precisely the intersection of two cliques, as follows.

[] A set is a minimal separator of a chordal graph if and only if there exist maximal cliques forming a separating pair such that . A chordal graph can also be characterized using clique trees as follows: [] A graph is chordal if and only if it has a clique tree. Let be a chordal graph and let be a clique tree of . The edges of can be labeled as the intersection of the endpoints, and these labels are exactly the minimal vertex separators. We denote by the multiset of labels of the edges of . A graph can have several distinct clique trees and when we are studying chordal graphs, we have the next result giving us that the multiset is independent of the clique tree.

[] Let be a chordal graph. The multiset of minimal vertex separators of is the same for every clique tree .

In the light of Theorem 1 from now on we omit the subscript and use simply .

A graph class is hereditary if, for every and every induced subgraph of , .

Our goal is to characterize hereditary subclasses of chordal graphs by the intersection and containment relations of their minimal vertex separators and by forbidden induced subgraphs.

As we shall see, not every restriction on the minimal vertex separators leads to a hereditary graph class. In order to obtain characterization by forbidden induced subgraphs we will impose the additional requirement of being hereditary.

## 2 Results

We start with an auxiliary result showing that minimal vertex separators are, in some sense, hereditary. Let be a chordal graph, be a minimal vertex separator of , be a proper subset of and . Then there exist cliques in such that separates and . Since is a minimal vertex separator of , there exist cliques in such that . Let be cliques in such that , . separates and , because if there exists a path in between a vertex and in , then there exists a path from a vertex of to a vertex of in , contradicting the fact that separates and in . Now suppose that is not minimal and let be a non-empty subset of such that separates . Then separates , contradicting the fact that is minimal vertex separator of .

In the following, we will consider the graphs depicted in Figure 1:

Let be a chordal graph. The following statements are equivalent:

• i) For every induced subgraph of and for every pair , ;

• ii) is -free.

Let be a graph satisfying and suppose that is not valid. Suppose that there exists an induced subgraph of isomorphic to with vertices and cliques , and . Then we have , and , a contradiction. Now suppose we have an induced subgraph of isomorphic to with vertices and and cliques , , and . Then we have , and , a contradiction. Hence is -free. Therefore .

Conversely, let be a graph satisfying and suppose that is not valid. Let be an induced subgraph of . Let be a clique tree of and be the multiset of minimal vertex separators of . Suppose that there exist , with . First, suppose that there exist adjacent edges with labels and let be cliques such that and . Suppose and . Since the cliques are maximal there must exist and . Then induces a . Now suppose that there exist non-adjacent edges with labels and let be cliques such that and . Without loss of generality we can consider that the path in from to contains and . Suppose and . Since the cliques are maximal we must have , and . If then is an induced of ; otherwise if then induces a ; else there exists . If then is ; if then is else , induce , a contradiction. Therefore .

Let be a chordal graph. The following statements are equivalent:

• i) For every induced subgraph of and for every pair , .

• ii) is claw-free.

Let be a graph satisfying and suppose that is not valid. Let be an induced subgraph of and let be an induced claw of , with cliques . Then we have , a contradiction. Hence is claw-free and .

Conversely let be a graph satisfying and suppose that is not valid. Let be an induced subgraph of . Let be a clique tree of , be the multiset of minimal vertex separators of and suppose that , with . First, suppose that there exist adjacent edges with labels and let be cliques such that and . Since the separators are equal, take . Since the cliques are maximal there must exist , and . Then induces a claw, a contradiction. Now suppose that adjacent with labels and let be distinct cliques such that and . Without loss of generality we can consider that every path in from to contains and . Let . Since the cliques are maximal we must have , and and then induces a claw, a contradiction. Hence .

Due to space restrictions the proofs of the following two lemmas will be omitted.

Let be a chordal graph. The following statements are equivalent:

• i) For every induced subgraph of and for every pair , is not strictly contained in ;

• ii) is dart-free.

Let be a chordal graph. The following statements are equivalent:

• i) For every induced subgraph of and for every pair , or ;

• ii) is (gem, butterfly)-free.

Let be a hereditary subclass of chordal graphs. Let , be a clique tree of and be the multiset of minimal vertex separators of . For each pair one of the following situations holds:

• (a) Disjunction: .

• (b) Equality: .

• (c) Containment: or .

• (d) Overlap: and .

Since we are interested in a hereditary class , we can note that, by Lemma 2, if we have Containment then we must allow Equality and if we have Overlap then we must allow Equality and Disjunction.

And again by hereditarity we can note that if a class is claw-free then it is dart-free; if it is -free then it is gem-free and if it is dart-free or -free then it is butterfly-free.

Hence all possible combinations of properties are characterized in the following theorem. Let be a chordal graph. Then for every induced subgraph of and for every pair , we have:

• (i) is (claw, gem)-free.

• (ii) is (, gem, butterfly)-free.

• (iii) or is (dart, gem)-free.

• (iv) or or or is (gem, butterfly)-free.

• (v) or or and is dart-free.

• (vi) or or is -free.

It follows from Lemmas 2-2 and Remark 2.

We remark that had been previously proved in   and  .

We now move our attention to Helly Property. Let be a family of subsets of a set . We say that satisfies the Helly property when every subfamily of consisting of pairwise intersecting subsets satisfies .

We say that a set is a witness that the Helly property does not hold if:

• , and

• .

Let be a chordal graph such that does not satisfy the Helly property and such that for every induced subgraph the family satisfies the Helly property. Let be a clique tree of and be the set of minimal vertex separators incident to leaves of . Then is a witness.

If there exist separators with then cannot be simultaneously in a witness. But then there exists witness or , wich contradicts the minimality of . Then suppose that for all pairs . Since is not a witness there exists and then is universal in and this implies that belongs to all minimal vertex separators of . But this implies that satisfies the Helly property, a contradiction.

Let be the hereditary class of chordal graphs such that , satisfies the Helly property. Then for any chordal graph , G is Hajós-free.

Let and let be an induced Hajós of , with cliques . Let . Then we have and , so does not satisfy the Helly property.

Conversely let and let be an induced subgraph of minimal in relation to the property that does not satisfy the Helly property and let be a clique tree of . By the previous Lemma, is a witness. Let . Then and such that . Without lost of generality, let . Note that if then . Indeed, suppose . Since is a witness, we know that . Then we have:

 ⎛⎝⋂X∈Sℓ(T′)∖{A,B,C}X⎞⎠∩A∩B∩C=∅⇒
 ⎛⎝⋂X∈Sℓ(T′)∖{A,B,C}X⎞⎠∩A∩B=∅⇒
 ⋂X∈Sℓ(T′)∖{C}X=∅.

But this implies that there exists a proper subset of that does not satisfy the Helly property, contradicting the minimality of . Then we can suppose that and . Let be the leaves of edges labeled by respectively in . Let be an exclusive vertex of . Then induces a Hajós.

## References

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