# Homotopy Type of Independence Complexes of Certain Families of Graphs

## TL;DR

This paper investigates the topological structure of independence complexes for specific graph families, showing they are homotopy equivalent to wedges of spheres or circles, and explores how graph perturbations affect this topology.

## Contribution

It characterizes the homotopy types of independence complexes for generalized Mycielskian and categorical product graphs, and analyzes the effects of graph perturbations.

## Key findings

- Independence complexes of generalized Mycielskian graphs are wedge sums of spheres.
- Independence complexes of categorical product graphs are wedge sums of circles.
- Perturbing a graph can make its independence complex homotopy equivalent to a suspension.

## Abstract

We show that the independence complexes of generalised Mycielskian of complete graphs are homotopy equivalent to a wedge sum of spheres, and determine the number of copies and the dimensions of these spheres. We also prove that the independence complexes of categorical product of complete graphs are wedge sum of circles, upto homotopy. Further, we show that if we perturb a graph $G$ in a certain way, then the independence complex of this new graph is homotopy equivalent to the suspension of the independence complex of $G$.