Chordal graphs, higher independence and vertex decomposable complexes

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