On the total Roman domination number of graphs

A Roman dominating function on a graph G is a function f : V (G) -> {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. A total Roman dominating function is a Roman dominating function with the additional property that the subgraph of G induced by the set of all vertices of positive weight has no isolated vertices. The weight of a total Roman dominating function f is the value Sigma(u is an element of V(G))f(u). The total Roman domination number of G, gamma(tR)(G), is the minimum weight of a total Roman dominating function in G. In this paper, we establish some sharp bounds on the total Roman domination number of a graph. In addition, we determine the total Roman domination number of grid graphs P-2 square P-n and P-3 square P-n for n >= 2.