The bondage number of chordal graphs

TL;DR

This paper investigates the bondage number in chordal graphs, establishing an upper bound related to the maximum clique size and demonstrating the bound's optimality.

Contribution

It proves that the bondage number of a chordal graph is at most its maximum clique size, providing a tight bound with proof of optimality.

Findings

Bondage number of chordal graphs is at most the size of their maximum clique.

The established bound is proven to be tight.

Provides theoretical insight into domination parameters of chordal graphs.

Abstract

A set $S\subseteq V(G)$ of a graph $G$ is a dominating set if each vertex has a neighbor in $S$ or belongs to $S$. Let $\gamma(G)$ be the cardinality of a minimum dominating set in $G$. The bondage number $b(G)$ of a graph $G$ is the smallest cardinality of a set edges $A\subseteq E(G)$ such that $\gamma(G-A)=\gamma(G)+1$. A chordal graph is a graph with no induced cycle of length four or more. In this paper, we prove that the bondage number of a chordal graph $G$ is at most the order of its maximum clique, that is, $b(G)\leq \omega(G)$. We show that this bound is best possible.