Generalized Bondage Number: The $k$-synchronous bondage number of a graph

TL;DR

This paper introduces the $k$-synchronous bondage number, a new graph metric measuring the minimum edges removal needed to increase the dominating number by $k$, with applications in network design.

Contribution

It defines the $k$-synchronous bondage number, analyzes it for various graph classes, and provides bounds, extending the concept of the traditional bondage number.

Findings

Determined $k$-synchronous bondage number for several graph classes.

Provided bounds for the $k$-synchronous bondage number in general graphs.

Suggested applications in network connectivity and optimization.

Abstract

We investigate a generalization of the bondage number of a graph called the \textit{$k\,$-synchronous bondage number}. The $k\,$-synchronous bondage number of a graph is the smallest number of edges that, when removed, increases the dominating number by $k$. In this paper, we discuss the 2-synchronous bondage number and then generalize to $k\,$-synchronous bondage number. We present $k\,$-synchronous bondage number for several graph classes and give bounds for general graphs. We propose this characteristic as a metric of the connectivity of a simple graph with possible uses in the field of network design and optimization.