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[t~]nsi = 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, O(G) = r(G) where T(G) is the maximum cardinality of a minimal dominating set of G and where O(G) is the maximum cardinality among all perfect neighborhood sets of G. @ 1999 Elsevier Science B.V. All rights reserved