Bounds on the disjunctive domination number of a tree

Abstract

A set D of vertices in a graph G is a disjunctive dominating set in G if every vertex not in D is adjacent to a vertex of D or has at least two vertices in D at distance 2 from it in G. The disjunctive domination number, γd2(G), of G is the minimum cardinality of a disjunctive dominating set in G. We show that if T is a tree of order n with l leaves and s support vertices, then n-l+3/4≤γd2(T)≤n+l+s/4. Moreover, we characterize the families of trees which attain these bounds.