Domination versus disjunctive domination in trees

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, γ (G), of G is the minimum cardinality of a dominating set of G. A set S of vertices in G is a disjunctive dominating set in G if every vertex not in S is adjacent to a vertex of S or has at least two vertices in S at distance 2 from it in G. The disjunctive domination number, γ₂^d (G), of G is the minimum cardinality of a disjunctive dominating set in G. It is known that there exist graphs G that belong to the class of bipartite graphs or claw-free graphs or chordal graphs such that γ(G) > C γ₂^d(G) for any given constant C. In this paper, we show that if T is a tree, then γ (T) < 2γ₂^d(T) - 1, and we provide a constructive characterization of the trees achieving equality in this bound.