Independence Complex of the Lexicographic Product of a Forest

TL;DR

This paper investigates the topological structure of independence complexes of lexicographic product graphs involving forests and other graphs, establishing conditions under which these complexes are homotopy equivalent to wedge sums of spheres and providing explicit calculations.

Contribution

It proves that for certain forests, the independence complex of their lexicographic product with a graph inherits the homotopy type of a wedge sum of spheres, extending known results.

Findings

Independence complex of $G[H]$ is a wedge of spheres if $G$ is a forest not dominated by a single vertex and $H$'s independence complex is a wedge of spheres.

Explicit calculation of the homotopy type of $L_m [H]$ when $H$'s independence complex is a wedge of spheres.

Description of the homological connectivity of $G[K]$ in terms of the independent domination number of $G$.

Abstract

We study the independence complex of the lexicographic product $G[H]$ of a forest $G$ and a graph $H$. We prove that for a forest $G$ which is not dominated by a single vertex, if the independence complex of $H$ is homotopy equivalent to a wedge sum of spheres, then so is the independence complex of $G[H]$. We offer two examples of explicit calculations. As the first example, we determine the homotopy type of the independence complex of $L_m [H]$, where $L_m$ is the tree on $m$ vertices with no branches, for any positive integer $m$ when the independence complex of $H$ is homotopy equivalent to a wedge sum of $n$ copies of $d$-dimensional sphere. As the second one, for a forest $G$ and a complete graph $K$, we describe the homological connectivity of the independence complex of $G[K]$ by the independent domination number of $G$.