Further results on secure restrained domination in graphs

Let G = (V, E) be a graph and let S subset of V. The set S is a dominating set of G if every vertex in V \ S is adjacent to some vertex in S. The set S is a restrained dominating set if every vertex in V \ S is adjacent to a vertex in S and to a vertex in V \ S. The minimum cardinality of a restrained dominating set is called restrained domination number of G and it is denoted by gamma(r) (G). A set S subset of V(G) is called a secure (restrained) dominating set if S is (restrained) dominating and for all u is an element of V \ S there exists v is an element of S boolean AND N(u) such that (S \ {v}) boolean OR {u} is (restrained) dominating. The minimum cardinality of a secure (restrained) dominating set is a secure (restrained) domination number of G and it is denoted by gamma(s) (G) (gamma(sr) (G)). In this paper we characterize few classes of graphs for which gamma(r) (G) = gamma(sr) (G) and gamma(s) (G) = gamma(sr) (G). Specific values of certain graphs are determined and Nordhaus Gaddum type results are discussed.