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.