Constructive characterizations concerning weak Roman domination in trees

Given a graph G, we consider γr(G), γ{R2}(G), γr2(G) and γR(G) as the weak Roman domination number, the Roman {2}-domination number, the 2-rainbow domination number and the Roman domination number of G, respectively. It is known that γr(G)≤γ{R2}(G)≤γr2(G)≤γR(G) holds for any graph G. In connection with this, constructive characterizations of the trees T that satisfy the equalities above that are related with the weak Roman domination number are given in this work. That is, the trees T for which γr(T)=γ{R2}(T), γr(T)=γr2(T) and γr(T)=γR(T) are described.