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, is the minimum cardinality of a dominating P-set. In this paper we present results on changing and unchanging of the domination number with respect to the nondegenerate and hereditary properties when a graph is modified by adding an edge or deleting a vertex.
In this paper we present results on changing and unchanging of the domination number with respect to the nondegenerate property P, denoted by γP (G), when a graph G is modified by deleting a vertex or deleting edges. A graph G is (γP (G), k)P -critical if γP (G − S) < γP (G) for any set S ( V (G) with |S| = k. Properties of (γP , k)P -critical graphs are studied. The plus bondage number with respect to the property P, denoted b + P (G), is the cardinality of the smallest set of edges U ⊆ E(G) such that γP (G − U) > γP (G). Some known results for ordinary domination and bondage numbers are extended to γP (G) and b + P (G). Conjectures concerning b + P (G) are posed.