a b s t r a c tA 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, γ d 2 (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 γ d 2 (G) for any given constant C . In this paper, we show that if T is a tree, then γ (T ) ≤ 2γ d 2 (T )−1, and we provide a constructive characterization of the trees achieving equality in this bound.
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