Independence complexes of generalized Mycielskian graphs

TL;DR

This paper investigates the topological properties of independence complexes of generalized Mycielskian graphs, revealing how their homotopy types relate to those of the original graph and its Kronecker double cover, with applications to specific graph classes.

Contribution

It establishes a relationship between the homotopy types of independence complexes of generalized Mycielskian graphs and their Kronecker double covers, providing explicit calculations for certain graph families.

Findings

Homotopy type of independence complex depends on original graph and its Kronecker double cover.

Explicit homotopy types calculated for paths, cycles, and categorical products of complete graphs.

Provides a method to determine topological properties of complex graph constructions.

Abstract

We show that the homotopy type of the independence complex of the generalized Mycielskian of a graph $G$ is determined by the homotopy type of the independence complex of $G$ and the homotopy type of the independence complex of the Kronecker double cover of $G$. As an application we calculate the homotopy type for paths, cycles and the categorical product of two complete graphs.