On vertex decomposability and regularity of graphs

TL;DR

This paper addresses questions about the relationship between vertex decomposability and Castelnuovo-Mumford regularity in simple graphs, providing counterexamples that disprove previous conjectures.

Contribution

It offers the first known counterexamples to two open questions regarding the properties of vertex decomposable graphs and their regularity.

Findings

Counterexamples disprove previous conjectures

Vertex decomposability does not imply certain regularity properties

Questions about graph regularity and decomposability are answered negatively

Abstract

There are two motivation questions in \cite{MTS, MTS1} about Castelnuovo-Mumford regularity and vertex decomposable of simple graph $G$. In this paper, we disprove the questions by providing of two counterexamples.