Chordal graphs, higher independence and vertex decomposable complexes.

Given a finite simple undirected graph G there is a simplicial complex Ind(G), called the independence complex, whose faces correspond to the independent sets of G. 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 r ≥ 1 , a subset of the vertex set is called r-independent if the connected components of the induced subgraph have cardinality at most r. The collection of all r-independent subsets of G form a simplicial complex called the r-independence complex and is denoted by Indr(G). It is known that when G is a chordal graph the complex Indr(G) 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 r-independence complex is vertex decomposable for all values of r. Further, for any r ≥ 2 we construct chordal graphs on 2 r + 2 vertices such that their r-independence complexes are not sequentially Cohen–Macaulay. [ABSTRACT FROM AUTHOR]