Independence complexes of chordal graphs

A Note on Independence Complexes of Chordal Graphs and Dismantling

“…Independence complexes of chordal graphs have received due attention in the literature [10,9,8,15,20,22,6]. They are vertex-decomposable [22], hence homotopy equivalent to wedges of spheres or to a point.…”

Algorithmic complexity of finding cross-cycles in flag complexes

“…In particular, they are vertex-decomposable [23] and therefore homotopy equivalent to wedges of spheres (also proved in [22,16]). Moreover, every wedge of spheres arises, up to homotopy, as an independence complex of a chordal graph [16]. Another reason to study chordal graphs in this context is that for this family of graphs cross-cycles detect all of the homology of the independence complex.…”

“…First, note that for an arbitrary graph G the problems of deciding if I(G) is simply-connected or contractible are both undecidable [6]. Moreover all previous work on topological features of chordal graphs [23,5,16,22] made use exclusively of the existence of simplicial vertices (see Section 2). Our algorithm in Theorem 5 makes essential use of the geometric intersection representation of chordal graphs and their tree models.…”

“…This allows us to study special cases. Unfortunately, not a lot has been done in this direction besides the recent polynomial time algorithm for forests [16].…”

“…This allows us to study special cases. Unfortunately, not a lot has been done in this direction besides the recent polynomial time algorithm for forests [16].In this contribution, we focus on the more general class of chordal graphs. This is a natural choice for the problem as the independence complexes of chordal graphs are quite well understood, namely they are wedges of spheres up to homotopy, and any wedge of spheres can be realized as the independence complex of a chordal graph, up to homotopy.…”