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 is an element of V - S satisfy N(x) boolean AND S not equal N(y) boolean AND S. The locating-domination number gamma(L)(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] boolean AND S not equal N[v] boolean AND S. The minimum cardinality of a differentiating-total dominating set of G is the differentiating-total domination number of G, denoted by gamma(D)(t)(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.