The bondage numbers and efficient dominations of vertex-transitive graphs

The bondage number of a graph G is the minimum number of edges whose removal results in a graph with larger domination number. A dominating set D is called an efficient dominating set of G   if |N-[v]∩D|=1|N-[v]∩D|=1 for every vertex v∈V(G)v∈V(G). In this paper we establish a tight lower bound for the bondage number of a vertex-transitive graph. We also obtain upper bounds for regular graphs by investigating the relation between the bondage number and the efficient domination. As applications, we determine the bondage number for some circulant graphs and tori by characterizing the existence of efficient dominating sets in these graphs.