Constructive characterizations of (γ_p, γ)- and (γ_p, γ_pr)-trees

Let G = (V, E) be a graph without isolated vertices. A set S ⊆ V is a domination set of G if every vertex in V - S is adjacent to a vertex in S, that is N [S] = V. The domination number of G, denoted by γ(G), is the minimum cardinality of a domination set of G. A set S ⊆ V is a paired-domination set of G if S is a domination set of G and the induced subgraph G [S] has a perfect matching. The paired-domination number, denoted by γ_pr(G), is defined to be the minimum cardinality of a paired-domination set S in G. A subset S ⊆ V is a power domination set of G if all vertices of V can be observed recursively by the following rules: (i) all vertices in N[S] are observed initially, and (ii) if an observed vertex u has all neighbors observed except one neighbor v, then v is observed (by u). The power domination number, denoted by γ_p(G), is the minimum cardinality of a power domination set of G. In this paper, the constructive characterizations for trees with γ_p = γ and γ_pr = γ_p are provided respectively.