Characterizations of trees with equal domination parameters

Lei G = (V, E) be a graph. A set S ⊆ V is a restrained dominating set, if every vertex not in S is adjacent to a vertex in S and to a vertex in V - S The restrained domination number of G, denoted by γ_r(G), is the minimum cardinality of a restrained dominating set of G. A set S ⊆ V is a weak dominating set of G if, for every u in V - S, there exists a v ∈ S such that uv ∈ E and deg u ≥ deg v. The weak domination number of G, denoted by γ_w(G) is the minimum cardinality of a weak dominating set of G. In this article we provide a constructive characterization of those trees with equal independent domination and restrained domination numbers. A constructive characterization of those trees with equal independent domination and weak domination numbers is also obtained.