Romeo C. Alota, Enrico L. Enriquez
Let G be a connected simple graph. A set S ⊆ V (G) is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V (G) \ S. The restrained domination number of G, denoted by γr(G), is the minimum cardinality of a restrained dominating set of G. Let D be a minimum restrained dominating set of G. A restrained dominating set S ⊆ (V (G) \ D) is called an inverse restrained dominating set of G with respect to D. The inverse restrained domination number of G denoted by γr−1 (G) is the minimum cardinality of an inverse restrained dominating set of G. An inverse restrained dominating set of cardinality γr−1 (G) is called γr−1 -set. A disjoint restrained dominating set of G is the set C = D ∪ S ⊆ V (G). The disjoint restrained domination number of G denoted by γγr(G) is the minimum cardinality of a disjoint restrained dominating set of G. A disjoint restrained dominating set of cardinality γγr(G) is called γγr -set. In this paper, we show that every integers k and n with 2 ≤ k ≤ n is realizable as disjoint restrained domination number, and order of G respectively. Further, we give the characterization of the disjoint restrained dominating set and give some important results. © Research India Publications.
Department of Mathematics, School of Arts and Sciences, University of San Carlos, Cebu City, 6000, Philippines
