  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.
