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

We investigate a generalization of the bondage number of a graph called the $k\mathit{\text{-synchronous}}\mathit{\text{bondage}}\mathit{\text{number}}$. 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.