Vertex decomposable graphs and obstructions to shellability

Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the 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.


Introduction
Let G = (V, E) be a graph with vertex set V = {x 1 , . . . , x n }. The independence complex of G, denoted I(G), is the simplicial complex with vertex set V and with faces the independent sets of G. When it causes no confusion, we will say that G satisfies some property if its independence complex does. For example, we will say that G is shellable if I(G) is shellable. The independence complex has been previously studied in e.g. [1,17,19].
The Stanley-Reisner ring of I(G) is k[x 1 , . . . , x n ]/(x i x j : x i x j ∈ E).
The quotient in the above ring is also called the edge ideal of G 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 C 3 and C 5 . In Section 4, we prove the main theorem of the paper: Theorem 1. (Main Theorem) If G is a graph with no chordless cycles of length other than 3 or 5, then G 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 combinatorics 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 G is the independence complex of the complement graph of G.
1.1. Cohen-Macaulay complexes. We review briefly the background definitions from geometric combinatorics and graph theory.
A simplicial complex ∆ is pure if all of its facets (maximal faces) are of the same dimension. A complex ∆ 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 F in ∆ is Let k be a field or the ring of integers. A complex ∆ is Cohen-Macaulay over k ifH i (link ∆ F ; k) = 0 for all faces F and i < dim(link ∆ F ). 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 k. Our results will be independent of the choice of k, 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 i-skeleton of ∆ is the complex generated by all the i-dimensional faces of ∆. A complex is sequentially Cohen-Macaulay if the pure i-skeleton is Cohen-Macaulay for all i. Any shellable complex is sequentially Cohen-Macaulay.
∆ is a Cohen-Macaulay complex if and only if the Stanley-Reisner ring of ∆ 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 n in a graph G is a path v 1 , v 2 , . . . , v n in G with no chord, i.e. with no edge v i v j with j = i + 1. Equivalently, the induced graph on {v 1 , . . . , v n } is the path on n vertices. In a like manner, a chordless cycle of length n is an induced n-cycle.
A graph is k-chordal if it has no chordless cycles of length > k, and chordal if it is 3-chordal.

Vertex decomposability and shedding vertices
A simplicial complex ∆ is recursively defined to be vertex decomposable if it is either a simplex or else has some vertex v so that (1) both ∆ \ v and link ∆ v are vertex decomposable, and (2) no face of link ∆ v is a facet of ∆ \ v. A vertex v which satisfies Condition (2) is called a shedding vertex. Vertex decompositions were introduced in the pure case by Provan and Billera [20] and extended to non-pure complexes by Björner and Wachs [5,Section 11].
A vertex decomposable complex is shellable. One proof of this fact is via the following lemma of independent interest: The shelling order in Lemma 3 is that of ∆ \ v, followed by the facets of v * link ∆ v in the order of the shelling of link ∆ v.
Both implications are known to be strict.
The definition of vertex decomposable (and so of shedding vertex) translates nicely to independence complexes. Let N (v) denote the open neighborhood of v, that is, all vertices adjacent to v. Proof. Translate the definitions! A shedding vertex of G is any vertex which satisfies Condition (2) of Lemma 4. A useful equivalent condition for shedding vertices is: We make a first observation: (so that in particular v and w are adjacent), then w is a shedding vertex for G.
Proof. Since every neighbor of v is also a neighbor of w, there are no edges from v to any vertex of G \ N [w]. Thus, any v can be added to any independent set in G \ N [w] while preserving independence.
Recall that a simplicial vertex is a vertex v such that N [v] 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: (1) Any neighbor of a simplicial vertex is a shedding vertex for G. 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 G, i.e., for understanding the independence complex of the complement of G. Since v and w 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 Engström also examined vertex decomposability in graphs, independently and at about the same time [11,Section 4] as I 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.

Cyclic graphs
Corollary 7 (2) states that if G has no chordless cycles of length greater than 3, then it is vertex decomposable. Let C n be the cyclic graph on n vertices. We discuss a partial converse: We give a geometric proof here.
We start with a technical lemma: Lemma 11. Let n = 2r + 1, 0 < d < r. Let the d-dimensional complex ∆ d n be the complex with vertex set Z/nZ and with facets F i = {i, i+ 2, . . . , i+ 2d} Proof. Consider ∆ d n for d > 1. A facet F i has codimension 1 intersection with two other facets: F i−2 and F i+2 . Since d > 1, and since all codimension 1 faces of F i other than F i ∩F i−2 and F i ∩F i+2 are "free" (contained in a unique facet), we can collapse Since n is odd, repeatedly adding 2 to some i ∈ Z/nZ will cover all vertices; hence ∆ 1 n is the 1-complex C n ∼ = S 1 . Proof. (Of Theorem 10) The if direction is easy: the independence complex of C 3 is three disconnected vertices, while that of C 5 is C 5 as a 1-complex. Both are clearly vertex decomposable.
In the other direction, we show that the pure d-skeleton is not Cohen-Macaulay, where d is the top dimension of the complex (i.e., d = dim I(C n )). There are two cases, based on whether n is even or odd. (It may be helpful to look at Figures 3.1 and 3.2 while reading the following.) Case 1. n = 2r. Then the top-dimensional facets have dimension r − 1, and there are two of them: one with all even vertices, the other with all odd vertices. As the pure r − 1 skeleton is not even connected, it is certainly not Cohen-Macaulay.
Case 2. n = 2r + 1, where r ≥ 2. Then the top-dimensional facets once more have dimension r−1. All such facets are obtained by taking a sequence of r alternating vertices in C n , with one skip of 2 vertices. We see that the top-dimensional skeleton of I(C n ) is the complex ∆ r−1 n discussed in Lemma 11, and so homotopic to S 1 . Thus, the pure r−1 skeleton is Cohen-Macaulay only when r − 1 = 1, i.e., when n = 5.
Since every pure skeleton of a sequentially Cohen-Macaulay complex is Cohen-Macaulay, we have shown that C n is not sequentially Cohen-Macaulay (hence not shellable or vertex decomposable) for n = 3, 5.

Proof of main theorem
The previous two sections motivate the result of Theorem 1. In this section, we will give a proof.
A simplicial k-path in G is a chordless path v 1 , v 2 , . . . , v k which cannot be extended on both ends to a chordless path v 0 , v 1 , . . . , v k , v k+1 in G. Thus, a simplicial vertex is a simplicial 1-path.
Chvátal, Rusu, and Sritharan [9] proved a nice generalization of Dirac's Theorem to (k + 2)-chordal graphs using simplicial k-paths. The following lemma of theirs will allow us to use the 5-chordal structure of G. Note 15. From a geometric combinatorics point of view, it might make more sense to count edge length and have the above definition be a simplicial (k − 1)-path, so that a simplicial vertex would be a simplicial 0-path. However, to avoid confusion, I have kept the original, more graph-theoretic definition.
We also need to use the lack of chordless 4-cycles: Lemma 16. Let w 1 , v, w 2 be a simplicial 3-path which is not a subgraph of any chordless C 4 in G. Then v is a shedding vertex.
Proof. We first notice that, since there is no edge w 1 w 2 , that any z adjacent to both w 1 and w 2 must also be adjacent to v. Otherwise, w 1 , v, w 2 , z would be a chordless 4-cycle.
Suppose by contradiction that v is not a shedding vertex. Then by Lemma 4 and Condition 5, there is an independent set in G \ N [v] which contains a vertex z 1 adjacent to w 1 , and a vertex z 2 adjacent to w 2 . Since z 1 , is adjacent to both w 1 and w 2 , so z 1 is not adjacent to w 2 , and z 2 is not adjacent to w 1 . Since z 1 and z 2 are in an independent set, z 1 is not adjacent to z 2 .
Counting non-adjacent pairs of vertices, we have just shown that z 1 , w 1 , v, w 2 , z 2 is a chordless path, which contradicts the definition of simplicial 3-path.
Thus v is a shedding vertex, as desired.
Proof. (Of Theorem 1) If G is chordal, then G is vertex decomposable, as shown in Corollary 7. Otherwise, G has some chordless 5-cycle, hence a chordless 3-path, and by Lemma 14 a simplicial 3-path. Lemma 16 shows that the middle vertex of any simplicial 3-path in G is a shedding vertex, and so by induction G is vertex decomposable.

5.1.
Obstructions. An obstruction to shellability is a non-shellable complex, all of whose proper subcomplexes are shellable. Thus, any non-shellable complex must contain at least one obstruction to shellability, while a shellable complex may or may not contain some obstructions to shellability as proper subcomplexes.
The order complex of a poset is the simplicial complex with vertex set the elements of the poset and with face set the chains of comparable elements. Thus, the order complex of P is the independence complex of the incomparability graph on P , which puts an edge between two elements if they are incomparable.
The study of obstructions to shellability was initiated by Billera and Myers, with the following theorem: Equivalently, the unique obstruction to shellability in a poset is D. We note that the incomparability graph of D is C 4 . Gallai gave a forbidden subgraph characterization of incomparability graphs of posets in [14] (translated to English in [21,Chapter 3]; a more accessible version of the list is in [23,Chapter 3.2]). The forbidden subgraphs include C n for n ≥ 5. Thus, Theorem 17 follows from Corollary 7 (2). Wachs studied obstructions to shellability further in [27], where she asked about the obstructions to shellability in a flag complex. Theorem 1 gives a classification, which we summarize in the following theorem: Theorem 18. The obstructions to shellability in flag complexes are exactly the independence complexes of C n , where n = 4 or n ≥ 6.
Proof. By Theorem 1, any non-shellable graph G has an induced subgraph (hence subcomplex) isomorphic to such a C n . In Theorem 10 we showed that such C n are not shellable, but any proper induced subgraph of C n is chordal, hence shellable.
A natural question suggested by Theorem 18 is whether there is some similar characterization of obstructions to shellability in non-flag complexes, where the minimal non-faces form a hypergraph. One might be led to ask whether the hypergraph of minimal non-faces is always cyclic in an obstruction to shellability. However, examples studied by Wachs [27] show this is not so, as follows.
Let We generalize this result in two respects. Let i(G) be the independent domination number, that is, the minimum cardinality of a maximal independent set. Any maximal independent set is a minimal dominating set, so γ(G) ≤ i(G).
Since a sequentially Cohen-Macaulay complex has homology vanishing below the dimension of the smallest facet, and since the smallest facet of I(G) has cardinality i(G), an immediate consequence is the following: Corollary 19. If G is any sequentially Cohen-Macaulay graph (over k), thenH j (I(G); k) = 0 for any j < i(G) − 1.
In particular, we recover the result [19, Theorem 1.2 (iii)] for chordal graphs and γ(G) and extend it to a larger class of graphs and a larger graph invariant.

Graph constructions
We now give examples of how shedding vertices can be used to show that certain graph constructions respect shellability.
Lemma 20. If G = G 1∪ G 2 , then I(G) = I(G 1 ) * I(G 2 ), the join of simplicial complexes. Hence G is vertex decomposable, shellable and/or sequentially Cohen-Macaulay if and only if G 1 and G 2 are.
Proof. It is obvious from the definition that I(G) is the given join and that the join is vertex decomposable if and only if both I(G 1 ) and I(G 2 ) are. That the join of two complexes is shellable or sequentially Cohen-Macaulay if and only if both complexes are is well known and can be found for example in [28]. (Part of Lemma 20 can be found as [24,Lemma 2.4].) Example 21. Adding a single vertex to G via disjoint union forms a cone over I(G). Adding on the graph consisting of two vertices connected by an edge via disjoint union corresponds to taking the suspension of I(G). Thus, for example, the union of n disjoint edges is homotopic to S n−1 .
Francisco and Hà [12], following Villarreal [25, Theorem 2.2], define a whisker in a graph as a vertex of degree 1. A similar idea seems to be studied in the wider graph theory literature under the name of pendant. We will prefer the latter term here. In [12] and [24], it is shown that, speaking broadly, adding pendants to graphs has good properties for maintaining shellability and the sequentially Cohen-Macaulay property. Their construction essentially works because adding a pendant adds a simplicial vertex. We give an obvious generalization: Proposition 22. Let G 0 be a graph with a complete subgraph K, and let G be obtained from G 0 by adding a new vertex v with edges to all vertices of K.
(That is, let G be obtained from G 0 by "starring K".) Then any element of K is a shedding vertex in G; conversely, G is shellable (sequentially Cohen-Macaulay) only if G 0 \ K is.
Proof. Since N [v] = K ∪{v}, we have that v is a simplicial vertex; hence any neighbor is a shedding vertex (Corollary 7). For the converse statement, we recall that links in a shellable/sequentially Cohen-Macaulay complex have the same property and notice that link The "clique-starring" construction described in Proposition 22 adds a pendant when |K| = 1. Whatever the size of K, the construction adds a simplicial vertex to G. We now consider a construction analogous to a pendant which adds a 3-simplicial path.
Proposition 23. Let G 0 be a graph with a complete subgraph K, and let K 1 , K 2 be disjoint subgraphs of K. Let G be obtained from G 0 by adding new vertices w 1 , w 2 , and v, with w 1 adjacent to all vertices of K 1 , w 2 adjacent to all vertices of K 2 , and v adjacent to w 1 and w 2 . Then v is a shedding vertex of G. Conversely, G is shellable (sequentially Cohen-Macaulay) only if G 0 is.
Proof. By definition, the path w 1 , v, w 2 is 3-simplicial, while v is in no chordless 4-cycles because K 1 and K 2 are disjoint. Lemma 16 shows that v is a shedding vertex. Conversely, link I(G) v = G 0 , and any link in a shellable/sequentially Cohen-Macaulay complex has the same property.
We notice in passing that G \ v = G 0 ∪ {w 1 , w 2 } is formed from G 0 by performing the construction of Proposition 22 on K 1 and K 2 .
Another graph construction is that of twinning. If two vertices v and w have the same neighbors, i.e., if N (w) = N (v), then we say v and w are true twins if there is an edge vw, and false twins otherwise. There are corresponding graph constructions: add a new vertex w to G, together with edges to all neighbors of some v. The family of distance hereditary graphs can be defined as the graphs that can be built from a single vertex by adding pendants, true twins, and false twins [7,Chapter 11.6]; twins are also useful in proofs of the Perfect Graph Theorem [10, Chapter 5.5].
Proposition 24. If v and w are true twins, then v and w are shedding vertices.
Proof. We note that N [v] = N [w], and thus by Lemma 6 they are both shedding vertices.
False twins are never shedding vertices, since any maximal independent Example 25. The 4-cycle is obtained by adding a false twin of the middle vertex in a 3-path. Thus, adding a false twin to a sequentially Cohen-Macaulay graph can result in a non-sequentially Cohen-Macaulay graph.
One more family of graph operations which has frequently been studied is that of graph products. There are a large number of such operations, based on different rules for putting edges on the Cartesian product of the vertex set. We cannot examine all of them, but note that the commonly considered operations of direct product and Cartesian product of graphs do not respect shellability or the sequential Cohen-Macaulay property, for the Cartesian product of two edges (shellable) is a 4-cycle (not sequentially Cohen-Macaulay), while the direct product of an edge with a 3-cycle (both shellable) is the complete bipartite graph K 3,3 , which [24,Corollary 3.11] shows is not sequentially Cohen-Macaulay.

A comment on perfect graphs
Herzog, Hibi, and Zheng [16] point out that classifying sequentially Cohen-Macaulay graphs is likely an intractable problem. We recall their argument. If ∆ is a simplicial complex, then the order complex of the face lattice of ∆ is a flag complex, and it is sequentially Cohen-Macaulay if and only if ∆ is. (The order complex of the face lattice is the barycentric subdivision of ∆.) Herzog, Hibi and Zheng conclude that characterizing sequentially Cohen-Macaulay graphs is as difficult as characterizing all sequentially Cohen-Macaulay complexes. The closely related property of shellability is likely of a similar difficulty.
As we have seen, however, there are families of graphs in which classifying the sequentially Cohen-Macaulay members of the family is possible. That all chordal graphs are sequentially Cohen-Macaulay (Corollary 7) is an example of this type of classification, as is the recursive characterization of sequentially Cohen-Macaulay bipartite graphs in [24,Corollary 3.11]. Other families of graphs may also have interesting answers.
We notice that the argument of Herzog, Hibi, and Zheng can help indicate the families of graphs in which we can hope for such a classification. For example, a perfect graph is one where every induced subgraph has chromatic number equal to the size of its largest clique. The Strong Perfect Graph Theorem says that a graph G is perfect if and only if there are no chordless odd cycles of length ≥ 5 in either G or its complement. Another fundamental result is that the complement of a perfect graph is also perfect. See [21] for more information and references about perfect graphs. Both chordal graphs and bipartite graphs are perfect, and characterizing the shellability and/or sequential Cohen-Macaulay connectivity of their common super-family would seem like a reasonable aim.
Unfortunately for this aim, poset (in)comparability graphs are perfect, as can be proved either by the direct argument of coloring elements by their rank, or else from Gallai's previously mentioned characterization of poset comparability graphs, which lack odd cycles of length ≥ 5 [14]. Moreover, a complex is sequentially Cohen-Macaulay if and only if its face poset is sequentially Cohen-Macaulay. Thus, characterizing the sequentially Cohen-Macaulay perfect graphs is at least as hard as characterizing which complexes in general are sequentially Cohen-Macaulay.
Considering the intersection of a graph family F with the family of poset incomparability graphs is a recommended exercise before looking for shellings of graphs in F.