Holes and a chordal cut in a graph

TL;DR

This paper investigates the existence of chordal cuts in graphs, specifically focusing on $K_{2,2,2}$-free hole-edge-disjoint graphs, and establishes conditions under which such graphs have chordal cuts.

Contribution

It introduces the concept of chordal cuts in graphs and provides a condition for their existence in a specific class of graphs.

Findings

Graphs with no $K_{2,2,2}$-holes and edge-disjoint holes have chordal cuts under certain conditions.

The paper characterizes when such graphs contain chordal cuts.

Provides a new perspective on the structure of hole-edge-disjoint graphs.

Abstract

A set $X$ of vertices of a graph $G$ is called a {\em clique cut} of $G$ if the subgraph of $G$ induced by $X$ is a complete graph and the number of connected components of $G-X$ is greater than that of $G$. A clique cut $X$ of $G$ is called a {\em chordal cut} of $G$ if there exists a union $U$ of connected components of $G-X$ such that $G[U \cup X]$ is a chordal graph. In this paper, we consider the following problem: Given a graph $G$, does the graph have a chordal cut? We show that $K_{2,2,2}$-free hole-edge-disjoint graphs have chordal cuts if they satisfy a certain condition.