The bondage number of graphs: good and bad vertices

The domination number (G) of a graph G is the minimum number of vertices in a set D such that every vertex of the graph is either in D or is adjacent to a member of D. Any dominating set D of a graph G with jDj = (G) is called a -set of G. A vertex x of a graph G is called: (i) -good if x belongs to some -set and (ii) -bad if x belongs to no -set. The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater then (G). In this paper we present new sharp upper bounds for b(G) in terms of -good and -bad vertices of G.