Identifying Restrained Domination in Graphs

Let G be a connected simple graph. A subset S of V(G) is a dominating set of G if for every v ∈ V(G)∖S, there exists x∈S such that xv ∈ E(G). An identifying code S of a graph G is a dominating set S⊆V(G) such that for every v ∈ V(G), N_G [v]∩S is distinct. An identifying code of a graph G is an identifying restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V(G)∖S. Alternately, an identifying code of a graph S⊆V(G) is an identifying restrained dominating set if N[S]=V(G) and 〈V(G)∖S〉 is a subgraph without isolated vertices. The minimum cardinality of an identifying restrained dominating set of G, denoted by γ_r^ID (G), is called the identifying restrained domination number of G. In this paper, we initiate the study of the concept and give the domination number of some special graphs. Further, we show the characterization of the identifying restrained dominating set in the join of two nontrivial connected graphs.