Restrained bondage in graphs

Let $G=(V,E)$ be a graph. A set $S?V$ is a restrained dominating set if every vertex not in $S$ is adjacent to a vertex in $S$ and to a vertex in $V?S$. The restrained domination number of $G$, denoted by ${\gamma }_r\left(G\right)$, is the smallest cardinality of a restrained dominating set of $G$. We define the restrained bondage number $b_r\left(G\right)$ of a nonempty graph $G$ to be the minimum cardinality among all sets of edges $E^{\prime }?E$ for which ${\gamma }_r(G?E^{\prime })>{\gamma }_r\left(G\right)$. Sharp bounds are obtained for $b_r\left(G\right)$, and exact values are determined for several classes of graphs. Also, we show that the decision problem for $b_r\left(G\right)$ is NP-complete even for bipartite graphs.