Shellability of Polyhedral Joins of Simplicial Complexes and Its Application to Graph Theory

TL;DR

This paper studies the shellability of polyhedral joins of simplicial complexes, providing conditions for shellability and applying these results to generalized lexicographic products of graphs to analyze their independence complexes.

Contribution

It introduces new conditions for shellability of polyhedral joins and applies these findings to graph theory, particularly in analyzing independence complexes of generalized lexicographic products.

Findings

Certain pairs (K, L) guarantee shellability regardless of M's shellability

Provided necessary and sufficient conditions for shellability of polyhedral joins

Applied results to specific graph constructions involving generalized lexicographic products

Abstract

We investigate the shellability of the polyhedral join $\mathcal{Z}^*_M (K, L)$ of simplicial complexes $K, M$ and a subcomplex $L \subset K$. We give sufficient conditions and necessary conditions on $(K, L)$ for $\mathcal{Z}^*_M (K, L)$ being shellable. In particular, we show that for some pairs $(K, L)$, $\mathcal{Z}^*_M (K, L)$ becomes shellable regardless of whether $M$ is shellable or not. Polyhedral joins can be applied to graph theory as the independence complex of a certain generalized version of lexicographic products of graphs which we define in this paper. The graph obtained from two graphs $G, H$ by attaching one copy of $H$ to each vertex of $G$ is a special case of this generalized lexicographic product and we give a result on the shellability of the independence complex of this graph by applying the above results.