On perfect neighborhood sets in graphs

Let G = (V, E) be a graph and let S ⊆ V.. The set S is a dominating set of G is every vertex of V − S is adjacent to a vertex of S. A vertex v of G is called S-perfect if |N[ν]∩S| = 1 where N[v] denotes the closed neighborhood of v. The set S is defined to be a perfect neighborhood set of G if every vertex of G is S-perfect or adjacent with an S-perfect vertex. We prove that for all graphs G, Θ(G) = Γ(G) where Γ(G) is the maximum cardinality of a minimal dominating set of G and where Θ(G) is the maximum cardinality among all perfect neighborhood sets of G.