A characterization of trees with equal Roman {2}-domination and Roman domination numbers

Given a graph G=(V,E) and a vertex v∈V, by N(v) we represent the open neighbourhood of v. Let f:V→{0,1,2} be a function on G. The weight of f is ω(f)=∑v∈Vf(v) and let Vi={v∈V:f(v)=i}, for i=0,1,2. The function f is said to be 1) a Roman {2}-dominating function, if for every vertex v∈V0, ∑u∈N(v)f(u)≥2. The Roman {2}-domination number, denoted by γ{R2}(G), is the minimum weight among all Roman {2}-dominating functions on G; 2) a Roman dominating function, if for every vertex v∈V0 there exists u∈N(v)∩V2. The Roman domination number, denoted by γR(G), is the minimum weight among all Roman dominating functions on G. It is known that for any graph G, γ{R2}(G)≤γR(G). In this paper, we characterize the trees T that satisfy the equality above.