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.