Distance r-domination number and r-independence complexes of graphs

For r >= 1, the r-independence complex of a graph G, denoted Ind(r)(G), is a simplicial complex whose faces are subsets I subset of & nbsp; V(G) such that each component of the induced subgraph G[I] has at most r vertices. In this article, we establish a relation between the distance r-domination number of G and (homological) connectivity of Ind(r)G). We also prove that Ind(r)(G), for a chordal graph G, is either contractible or homotopy equivalent to a wedge of spheres. Given a wedge of spheres, we also provide a construction of a chordal graph whose r-independence complex has the homotopy type of the given wedge. (C)& nbsp;2022 Elsevier Ltd. All rights reserved.