Shellability, vertex decomposability, and lexicographical products of graphs

TL;DR

This paper studies the conditions under which the independence complex of lexicographical graph products is shellable or vertex decomposable, and constructs an infinite family of graphs with shellable but not vertex decomposable independence complexes.

Contribution

It characterizes when the independence complex of lexicographical graph products is shellable or vertex decomposable and provides a new family of graphs with specific topological properties.

Findings

Identifies conditions for shellability and vertex decomposability of independence complexes

Constructs an infinite family of graphs with shellable but not vertex decomposable complexes

Advances understanding of topological properties of graph products

Abstract

We investigate when the independence complex of $G[H]$, the lexicographical product of two graphs $G$ and $H$, is either vertex decomposable or shellable. As an application, we construct an infinite family of graphs with the property that every graph in this family has the property that the independence complex of each graph is shellable, but not vertex decomposable.