Higher Independence Complexes of graphs and their homotopy types

Priyavrat Deshpande; Anurag Singh

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arXiv · 2001.05448

Higher Independence Complexes of graphs and their homotopy types

Priyavrat Deshpande·Anurag Singh

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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.

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Priyavrat Deshpande, Anurag Singh. 2020-01-15. Higher Independence Complexes of graphs and their homotopy types. https://arxiv.org/abs/2001.05448

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