On independence complexes of graph products

TL;DR

This paper investigates the topological properties of independence complexes in graph products involving paths, determining their homotopy types and analyzing subgraph complexes to advance understanding in algebraic combinatorics.

Contribution

It provides explicit homotopy type results for independence complexes of specific graph products and their induced subgraphs, expanding the theoretical framework of combinatorial topology.

Findings

Homotopy types of independence complexes for P_n×P_m, P_n⊠P_2, P_n⊠P_3, P_n⊠P_4

Analysis of independence complexes of induced subgraphs of these products

Results on lexicographic product complexes G∘H

Abstract

We study the independence complexes of graph products where at least one factor is a path. We also analyze the complexes of their induced subgraphs. We determine the homotopy type of the independence complex of the graphs $P_n\times P_m$, $P_n\boxtimes P_2$, $P_n\boxtimes P_3$ and $P_n\boxtimes P_4$. We also focus in the independence complexes of the induced subgraphs of $P_n\times P_3$, $P_n\boxtimes P_2$, $P_n\boxtimes P_3$, $P_n\boxtimes P_4$ and some lexicographic products $G\circ H$.