On locating and differetiating-total domination in trees

A total dominating set of a graph G = (V, E) with no isolated vertex is a set S ⊆ V such that every vertex is adjacent to a vertex in S. A total dominating set S of a graph G is a locating-total dominating set if for every pair of distinct vertices u and v in V − S, N(u) ∩ S 6= N(v) ∩ S, and S is a differentiating-total dominating set if for every pair of distinct vertices u and v in V , N [u]∩ S 6= N [v]∩ S. Let γ t (G) and γ t (G) be the minimum cardinality of a locating-total dominating set and a differentiating-total dominating set of G, respectively. We show that for a nontrivial tree T of order n, with ` leaves and s support vertices, γ t (T ) > max{2(n+ `− s+1)/5, (n+2− s)/2}, and for a tree of order n ≥ 3, γ t (T ) ≥ 3(n+`−s+1)/7, improving the lower bounds of Haynes, Henning and Howard. Moreover we characterize the trees satisfying γ t (T ) = 2(n + ` − s + 1)/5 or γ t (T ) = 3(n + ` − s + 1)/7.