Simple homotopy types of independence complexes of graphs involving grid graphs

TL;DR

This paper investigates the homotopy types of independence complexes of graphs containing grid subgraphs, generalizing previous results and providing explicit homotopy type descriptions for certain grid graphs.

Contribution

It generalizes Csorba's result by relating grid subgraphs to the homotopy types of independence complexes, offering new insights into their topological structure.

Findings

Independence complexes of certain grid graphs are homotopy equivalent to simplicial suspensions.

The paper determines the simple homotopy types of independence complexes for specific grid graphs.

A new method relates grid subgraphs to the topological properties of independence complexes.

Abstract

We show that if a graph $G$ involves a certain square grid graph as a full subgraph, then a certain operation on it yields a simplicial suspension of the independence complex of $G$. This generalizes a result of Csorba. As a corollary, we determine the simple homotopy types of the independence complexes of some grid graphs.