Higher Independence Complexes of graphs and their homotopy types

TL;DR

This paper determines the homotopy types of r-independence complexes for various graph families, showing they are either wedges of spheres or contractible, and provides explicit formulas for these types.

Contribution

It explicitly computes the homotopy types of r-independence complexes for specific graph classes, extending previous understanding of their topological structure.

Findings

Homotopy types are either wedges of spheres or contractible.

Provides closed-form formulas for the homotopy types.

Analyzes complexes for complete s-partite, whiskered, cycle, and perfect m-ary trees.

Abstract

For $r\geq 1$, the $r$-independence complex of a graph $G$ is a simplicial complex whose faces are subset $I \subseteq V(G)$ such that each component of the induced subgraph $G[I]$ has at most $r$ vertices. In this article, we determine the homotopy type of $r$-independence complexes of certain families of graphs including complete $s$-partite graphs, fully whiskered graphs, cycle graphs and perfect $m$-ary trees. In each case, these complexes are either homotopic to a wedge of equi-dimensional spheres or are contractible. We also give a closed form formula for their homotopy types.