Splittings of independence complexes and the powers of cycles

TL;DR

This paper investigates when independence complexes of graphs can be decomposed into simpler parts, providing a recursive method to understand the homotopy types of powers of cycles, thus solving an open problem.

Contribution

It introduces a recursive approach to determine the homotopy types of independence complexes of cycle powers, expanding understanding of their topological structure.

Findings

Identifies conditions for independence complex splittings

Provides a recursive relation for cycle powers

Answers an open question by D. Kozlov

Abstract

We use two cofibre sequences to identify some combinatorial situations when the independence complex of a graph splits into a wedge sum of smaller independence complexes. Our main application is to give a recursive relation for the homotopy types of the independence complexes of powers of cycles, which answers an open question of D. Kozlov.