The complement of a connected bipartite graph is vertex decomposable

TL;DR

This paper proves that the complement of a connected bipartite graph is vertex decomposable, which implies it has desirable algebraic properties like Cohen-Macaulayness, enriching the understanding of graph complements in combinatorial algebra.

Contribution

It establishes that the complement of any connected bipartite graph is vertex decomposable, a property not previously characterized for this class of graphs.

Findings

Complement of connected bipartite graphs is vertex decomposable.

Vertex decomposability implies Cohen-Macaulayness for these graph complements.

Advances understanding of algebraic properties of graph complements.

Abstract

Associated to a simple undirected graph $G$ is a simplicial complex $\Delta_G$ whose faces correspond to the independent sets of $G$. A graph $G$ is called vertex decomposable if $\Delta_G$ is a vertex decomposable simplicial complex. We are interested in determining what families of graph have the property that the complement of $G$, denoted by $\overline{G}$, is vertex decomposable. We obtain the result that the complement of a connected bipartite graph is vertex decomposable and so it is Cohen-Macaulay due to pureness of $\Delta_{\overline{G}}$.