Vander Meulen, Kevin N., Adam Van Tuyl. “Shellability, vertex decomposability, and lexicographical products of graphs.” Contributions to Discrete Mathematics 12, no. 2 (2017): 63-68.
Abstract
In this note we describe 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 show that there exists an infinite family of graphs whose independence complexes are shellable but not vertex decomposable.