ON THE TOTAL ROMAN DOMINATION IN TREES

A total Roman dominating function on a graph G is a function f : V(G) -> {0,1, 2} satisfying the following conditions: (i) every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f(v) = 2 and (ii) the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The weight of a total Roman dominating function f is the value f (V (G)) = Sigma(u is an element of V(G))f(u). The total Roman domination number gamma(tR) (G) is the minimum weight of a total Roman dominating function of G. Ahangar et al. in [H.A. Ahangar, M.A. Henning, V. Samodivkin and I.G. Yero, Total Roman domination in graphs, Appl. Anal. Discrete Math. 10 (2016) 501-517] recently showed that for any graph G without isolated vertices, 2 gamma(G) <= gamma(tR) (G) <= 3 gamma(G), where gamma(G) is the domination number of G, and they raised the problem of characterizing the graphs G achieving these upper and lower bounds. In this paper, we provide a constructive characterization of these trees.