Restrained bondage in graphs

Let G=(V,E)G=(V,E) be a graph. A set S⊆VS⊆V is a restrained dominating set if every vertex not in SS is adjacent to a vertex in SS and to a vertex in V−SV−S. The restrained domination number of GG, denoted by γr(G)γr(G), is the smallest cardinality of a restrained dominating set of GG. We define the restrained bondage number  br(G)br(G) of a nonempty graph GG to be the minimum cardinality among all sets of edges E′⊆EE′⊆E for which γr(G−E′)>γr(G)γr(G−E′)>γr(G). Sharp bounds are obtained for br(G)br(G), and exact values are determined for several classes of graphs. Also, we show that the decision problem for br(G)br(G) is NP-complete even for bipartite graphs.