LOCATING AND DIFERINTIAL 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 total dominating set S is a locating-total dominating set of G if every two vertices x, y 2 V (G) S satisfy N(x) \ S ̸ = N(y) \ S. The locating-total domination number γ t L (G) is the minimum cardinality of a locating-total 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 γ D t (G). We obtain new bounds for the Locating-domination number, and the differentiating- total domination number in trees. We improve previous bounds presented in [M. Chellali, On locating and differentiating-total domination in trees, Discussiones Math. Graph Theory 28(3) (2008), 383-392] and [X.-g. Chen, M. Y. Sohn, Bounds on the locating- total domination number of a tree, Discrete Appl. Math. 159(13-14)(2011), 769-773] for the locating-total domination number, and [W. Ning et al. Bounds on the differentiating- total domination number of a tree, Discrete Applied Mathematics (2016), In press] for the differentiating-total domination number. Moreover, we characterize all trees achieving equality for the new bounds.