Chordal graphs, higher independence and vertex decomposable complexes

Abstract

Given a finite simple undirected graph [Formula: see text] there is a simplicial complex [Formula: see text], called the independence complex, whose faces correspond to the independent sets of [Formula: see text]. This is a well-studied concept because it provides a fertile ground for interactions between commutative algebra, graph theory and algebraic topology. In this paper, we consider a generalization of independence complex. Given [Formula: see text], a subset of the vertex set is called [Formula: see text]-independent if the connected components of the induced subgraph have cardinality at most [Formula: see text]. The collection of all [Formula: see text]-independent subsets of [Formula: see text] form a simplicial complex called the [Formula: see text]-independence complex and is denoted by [Formula: see text]. It is known that when [Formula: see text] is a chordal graph the complex [Formula: see text] has the homotopy type of a wedge of spheres. Hence, it is natural to ask which of these complexes are shellable or even vertex decomposable. We prove, using Woodroofe’s chordal hypergraph notion, that these complexes are always shellable when the underlying chordal graph is a tree. Using the notion of vertex splittable ideals we show that for caterpillar graphs the associated [Formula: see text]-independence complex is vertex decomposable for all values of [Formula: see text]. Further, for any [Formula: see text] we construct chordal graphs on [Formula: see text] vertices such that their [Formula: see text]-independence complexes are not sequentially Cohen–Macaulay.