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VERTEX DECOMPOSABLE GRAPHS and OBSTRUCTIONS to SHELLABILITY 1. Introduction

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VERTEX DECOMPOSABLE GRAPHS and OBSTRUCTIONS to SHELLABILITY 1. Introduction PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 137, Number 10, October 2009, Pages 3235–3246 S 0002-9939(09)09981-X Article electronically published on June 4, 2009 VERTEX DECOMPOSABLE GRAPHS AND OBSTRUCTIONS TO SHELLABILITY RUSS WOODROOFE (Communicated by Jim Haglund) Abstract. Inspired by several recent papers on the edge ideal of a graph , we study the equivalent notion of the independence complex of .Usingthe tool of vertex decomposability from geometric combinatorics, we show that 5- chordal graphs with no chordless 4-cycles are shellable and sequentially Cohen- Macaulay. We use this result to characterize the obstructions to shellability in flag complexes, extending work of Billera, Myers, and Wachs. We also show how vertex decomposability may be used to show that certain graph constructions preserve shellability. 1. Introduction Let =(,) be a graph with vertex set = {1,...,}.Theindependence complex of , denoted (), is the simplicial complex with vertex set and with faces the independent sets of . When it causes no confusion, we will say that satisfies some property if its independence complex does. For example, we will say that is shellable if () is shellable. The independence complex has been previously studied in e.g. [1, 17, 19]. The Stanley-Reisner ring of ()is [1,...,]/( : ∈ ). The quotient in the above ring is also called the edge ideal of and has been an object of study in its own right [26]. In particular, a recent series of papers [12, 13, 16, 24] has worked from the edge ideal to show that chordal graphs are sequentially Cohen-Macaulay and shellable and that certain graph constructions preserve shellability and/or being sequentially Cohen-Macaulay. In this paper, we consider vertex decomposability in graphs. In Section 2, we recall the definition of a vertex decomposable simplicial complex and show what this means for (the independence complexes of) graphs. As an easy consequence we recover the result that chordal graphs are shellable, hence sequentially Cohen- Macaulay. In Section 3, we give a geometric proof that the only cyclic graphs which are vertex decomposable, shellable and/or sequentially Cohen-Macaulay are 3 and 5. In Section 4, we prove the main theorem of the paper: Received by the editors January 8, 2009. 2000 Mathematics Subject Classification. Primary 13F55, 05C38, 05E99. Key words and phrases. Sequentially Cohen-Macaulay, independence complex, edge ideal, chordal graphs. ⃝c 2009 American Mathematical Society Reverts to public domain 28 years from publication 3235 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3236 RUSS WOODROOFE Theorem 1 (Main Theorem). If is a graph with no chordless cycles of length other than 3 or 5,then is vertex decomposable (hence shellable and sequentially Cohen-Macaulay.) In Section 5, we reinterpret Theorem 1 in terms of obstructions to shellings, answering a question of Wachs. We also give an application to domination numbers, in the style of [19]. In Section 6, we examine several graph constructions that preserve vertex decomposability. Finally, in Section 7 we close with some comments on classes of sequentially Cohen-Macaulay graphs. Note 2. Independence complexes have been studied more extensively in the combi- natorics literature as flag complexes [22, Chapter III.4 and references]. Many papers on flag complexes study them by considering the clique complex. We notice that the clique complex of a graph is the independence complex of the complement graph of . 1.1. Cohen-Macaulay complexes. We review briefly the background definitions from geometric combinatorics and graph theory. A simplicial complex Δ is pure if all of its facets (maximal faces) are of the same dimension. A complex Δ is shellable if its facets fit together nicely. The precise definition will not be important to us, but can be found, with much additional background, in [28, Lecture 3]. The link of a face in Δ is linkΔ = { : ∪ is a face in Δ,∩ = ∅}. Let be a field or the ring of integers. A complex Δ is Cohen-Macaulay over ˜ if (linkΔ ; ) = 0 for all faces and <dim(linkΔ ). More intuitively, a complex is Cohen-Macaulay if it has the homology of a bouquet of top-dimensional spheres and if every link satisfies the same condition. It is a well-known fact that any Cohen-Macaulay complex is pure. Any pure, shellable complex is Cohen-Macaulay over any . Our results will be independent of the choice of , and we henceforth drop it from our notation. Since simplicial complexes that are not pure are often interesting, we study Stanley’s extension [22, Chapter III.2] of the definition of Cohen-Macaulay (and its relationship with shellability) to arbitrary simplicial complexes. The pure -skeleton of Δ is the complex generated by all the -dimensional faces of Δ. A complex is sequentially Cohen-Macaulay if the pure -skeleton is Cohen-Macaulay for all .Any shellable complex is sequentially Cohen-Macaulay. Δ is a Cohen-Macaulay complex if and only if the Stanley-Reisner ring of Δ is a Cohen-Macaulay ring. There is also a ring-theoretic notion of sequentially Cohen- Macaulay [22, Definition III.2.9]. For more background, refer to [4] and [22] for the combinatorial point of view or to [8] for a more ring-theoretic approach. 1.2. Chordless paths and cycles. A chordless path of length in a graph is apath1,2,..., in with no chord, i.e. with no edge with ∕= +1. Equivalently, the induced graph on {1,...,} is the path on vertices. In a like manner, a chordless cycle of length is an induced -cycle. Agraphis-chordal if it has no chordless cycles of length >,andchordal if it is 3-chordal. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use VERTEX DECOMPOSABLE GRAPHS 3237 2. Vertex decomposability and shedding vertices A simplicial complex Δ is recursively defined to be vertex decomposable if it is either a simplex or else has some vertex so that (1) both Δ ∖ and linkΔ are vertex decomposable, and (2) no face of linkΔ is a facet of Δ ∖ . Avertex which satisfies Condition (2) is called a shedding vertex. Vertex decom- positions were introduced in the pure case by Provan and Billera [20] and extended to non-pure complexes by Bj¨orner and Wachs [5, Section 11]. A vertex decomposable complex is shellable. One proof of this fact is via the following lemma of independent interest: Lemma 3 (Wachs [27, Lemma 6]). If Δ is a simplicial complex with shedding vertex and if both Δ ∖ and linkΔ are shellable, then Δ is shellable. The shelling order in Lemma 3 is that of Δ∖, followed by the facets of ∗linkΔ in the order of the shelling of linkΔ . To summarize, we have the chain of implications: vertex decomposable =⇒ shellable =⇒ sequentially Cohen-Macaulay. Both implications are known to be strict. The definition of vertex decomposable (and so of shedding vertex) translates nicely to independence complexes. Let ()denotetheopen neighborhood of , that is, all vertices adjacent to .Let[]denotetheclosed neighborhood of , which is () together with itself, so that []=() ∪{}. Lemma 4. An independence complex () is vertex decomposable if is a totally disconnected graph (with no edges) or if (1) ∖ and ∖ [] are both vertex decomposable, and (2) no independent set in ∖ [] is a maximal independent set in ∖ . Proof. Translate the definitions! □ A shedding vertex of is any vertex which satisfies Condition (2) of Lemma 4. A useful equivalent condition for shedding vertices is: Condition 5. For every independent set contained in ∖ [], there is some ∈ ()sothat ∪{} is independent. We make a first observation: Lemma 6. If [] ⊆ [] (so that in particular and are adjacent), then is a shedding vertex for . Proof. Since every neighbor of is also a neighbor of , there are no edges from to any vertex of ∖ []. Thus, any can be added to any independent set in ∖ [] while preserving independence. □ Recall that a simplicial vertex is a vertex such that [] is a clique. A well- known theorem of Dirac [21, Theorem 6.3] says that every chordal graph has a simplicial vertex. Thus, we have: Corollary 7. (1) Any neighbor of a simplicial vertex is a shedding vertex for . License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3238 RUSS WOODROOFE (2) A chordal graph is vertex decomposable (hence shellable [24, Theorem 1.2] and sequentially Cohen-Macaulay [13, Theorem 1.2]; also closely related is [19, Theorem 1.4]). Proof. If is a simplicial vertex and is a neighbor of ,then[] ⊆ [], and Lemma 6 gives (1). For (2), the theorem of Dirac thus says that a chordal graph has a shedding vertex if it is not totally disconnected. Since every induced subgraph of a chordal graph is chordal, both ∖ and ∖ [] are inductively vertex decomposable. □ Remark 8. Vertices satisfying the condition of Lemma 6 have been studied before under the name dominant vertices, in the context of so-called dismantlable graphs [15, 6]. However, dismantlability is a tool for understanding the homotopy type of the clique complex of , i.e., for understanding the independence complex of the complement of .Since and will not be adjacent in the complement, there does not seem to be any direct interpretation of dismantlability in terms of vertex decomposability. Remark 9. Anton Dochtermann and Alexander Engstr¨om also examined vertex decomposability in graphs, independently and at about the same time [11, Section 4] as the author did. In particular, they prove Corollary 7, and a special case of Proposition 22; they also notice that the result of Billera and Myers discussed in Section 5.1 is a special case of Corollary 7. 3. Cyclic graphs Corollary 7 (2) states that if has no chordless cycles of length greater than 3, then it is vertex decomposable.
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