Bounds on the locating-domination number and differentiating-total domination number in trees

A subset S of vertices in a graph G = (V,E) is a dominating set of G if every vertex in V − S has a neighbor in S, and is a total dominating set if every vertex in V has a neighbor in S. A dominating set S is a locating-dominating set of G if every two vertices x, y ∈ V − S satisfy N(x) ∩ S ≠ N(y) ∩ S. The locating-domination number γL(G) is the minimum cardinality of a locating-dominating set of G. A total dominating set S is called a differentiating-total dominating set if for every pair of distinct vertices u and v of G, N[u] ∩ S ≠ N[v] ∩ S. The minimum cardinality of a differentiating-total dominating set of G is the differentiating-total domination number of G, denoted by γtD(G) $\gamma _t^D (G)$ . We obtain new upper bounds for the locating-domination number, and the differentiating-total domination number in trees. Moreover, we characterize all trees achieving equality for the new bounds.