The bondage number of random graphs

TL;DR

This paper investigates the bondage number of binomial random graphs, providing bounds and a concentration result for the domination number, advancing understanding of graph resilience and domination properties in probabilistic models.

Contribution

It establishes a lower bound for the bondage number of G(n,p) that matches the trivial upper bound and offers a concentration result for the domination number.

Findings

Lower bound matches the order of the trivial upper bound

Provides a one-point concentration result for the domination number

Enhances understanding of domination properties in random graphs

Abstract

A dominating set of a graph is a subset $D$ of its vertices such that every vertex not in $D$ is adjacent to at least one member of $D$. The domination number of a graph $G$ is the number of vertices in a smallest dominating set of $G$. The bondage number of a nonempty graph $G$ is the size of a smallest set of edges whose removal from $G$ results in a graph with domination number greater than the domination number of $G$. In this note, we study the bondage number of binomial random graph $G(n,p)$. We obtain a lower bound that matches the order of the trivial upper bound. As a side product, we give a one-point concentration result for the domination number of $G(n,p)$ under certain restrictions.