a characterization relating domination, semitotal domination and total roman domination in trees

a total roman dominating function on a graph g is a function f:v(g)→{0,1,2} such that for every vertex v∈v (g) with f(v)=0 there exists a vertex u∈v(g) adjacent to v with f(u)=2, and the subgraph induced by the set {x∈v(g):f(x)≥1} has no isolated vertices. the total roman domination number of g, denoted γtr(g), is the minimum weight ω(f)=∑v∈v(g)f(v) among all total roman dominating functions f on g. it is known that γtr(g)≥γt2(g)+γ(g) for any graph g with neither isolated vertex nor components isomorphic to k2, where γt2(g) and γ(g) represent the semitotal domination number and the classical domination number, respectively. in this paper we give a constructive characterization of the trees that satisfy the equality above.