# Journal of the Ramanujan Mathematical Society

Volume 36, Issue 1, March 2021 pp. 53–71.

Higher independence complexes of graphs and their homotopy types

**Authors**:
Priyavrat Deshpande and Anurag Singh

**Author institution:**Chennai Mathematical Institute, India

**Summary: **
For r ≥ 1, the r-independence complex of a graph G is a simplicial complex whose faces are subsets
I ⊆ 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 equidimensional spheres or are contractible. We also give a closed form formula for their homotopy types.

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