Domination versus disjunctive domination in graphs

A dominating set in a graph G is a set S of vertices of G such that every vertex not in S is adjacent to a vertex of S. The domination number of G is the minimum cardinality of a dominating set of G. For a positive integer b, a set S of vertices in a graph G is a b-disjunctive dominating set in G if every vertex v not in S is adjacent to a vertex of S or has at least b vertices in S at distance 2 from it in G. The b-disjunctive domination number of G is the minimum cardinality of a b-disjunctive dominating set. In this paper, we continue the study of disjunctive domination in graphs. We present properties of b-disjunctive dominating sets in a graph. A characterization of minimal b-disjunctive dominating sets is given. We obtain bounds on the ratio of the domination number and the b-disjunctive domination number for various families of graphs, including regular graphs and trees.