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 is an element of 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 is an element of V(G) : f(x) >= 1} has no isolated vertices. The total Roman domination number of G, denoted gamma(tR)(G), is the minimum weight omega(f) = Sigma(v is an element of V(G))f(v) among all total Roman dominating functions f on G. It is known that gamma(tR)(G) >= gamma(t2)(G) + gamma(G) for any graph G with neither isolated vertex nor components isomorphic to K-2, where gamma(t2)(G) and gamma(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.