Total Roman domination and 2-independence in trees

bstract. Let G = (V, E) be a simple graph with vertex set V and edge set E. A total Roman dominating function on a graph G is a function f : V → {0, 1, 2} satisfying the following conditions:(i) every vertex u such that f(u) = 0 is adjacent to at least one vertex v such that 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 ) = Σu∈V (G)f(u). The total Roman domination number γtR(G) of G is the minimum weight of a total Roman dominating function of G. A subset S of V is a 2-independent set of G if every vertex of S has at most one neighbor in S. The maximum cardinality of a 2-independent set of G is the 2-independence number β2(G). These two parameters are incomparable in general, however, we show that if T is a tree, then γtR(T ) ≤ 3 β2(T )and we characterize all trees attaining the equality.