Domination with respect to nondegenerate properties: bondage number

For a graphical property P and a graph G, a subset S of vertices of G is a P-set if the subgraph induced by S has the property P . The domination number with respect to the property P , denoted by γP(G), is the minimum cardinality of a dominating P-set. The bondage number with respect to the property P of a nonempty graph G, denoted bP(G), is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number with respect to P unequal to γP(G). In this paper we show that some known sharp upper bounds for the ordinary bondage number are sharp upper bounds for bP(G) as well.