A note on the locating-total domination in trees

A total dominating set of a graph G = (V,E) with no isolated vertex is a set D ⊆ V (G) such that every vertex is adjacent to a vertex in D. A total dominating set D of G is a locating-total dominating set if for every pair of distinct vertices u and v in V −D, N(u) ∩D = N(v) ∩D. Let γ L(G) be the minimum cardinality of a locating-total dominating set of G. We show that for a nontrivial tree T of order n, with leaves and s support vertices, γ t (T ) ≥ (n + 2 − s + 1)/2, improving some previous bounds presented by Chellali [Discussiones Math. Graph Theory 28 (3) (2008), 383–392] and Chen and Young Sohn [Discrete Appl. Math. 159 (13-14) (2011), 769–773]. We also characterize the extremal trees achieving the above bound.