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 γr(G), is the smallest cardinality of a restrained dominating set of G. We define the restrained bondage number br(G) of a nonempty graph G to be the minimum cardinality among all sets of edges E′ ⊆ E for which γr(G−E) > γr(G). Sharp bounds are obtained for br(G), and exact values are determined for several classes of graphs. Also, we show that the decision problem for br(G) is NP-complete even for bipartite graphs.