Bounds on the locating-total domination number of a tree

In this paper, we continue the study of locating-total domination in graphs, introduced by Haynes et al. [T.W. Haynes, M.A. Henning, J. Howard, Locating and total dominating sets in trees, Discrete Applied Mathematics 154 (8) (2006) 1293–1300]. A total dominating set SS in a graph G=(V,E)G=(V,E) is a locating-total dominating set of GG if, for every pair of distinct vertices uu and vv in V−SV−S, NG(u)∩S≠NG(v)∩SNG(u)∩S≠NG(v)∩S. The minimum cardinality of a locating-total dominating set is the locating-total domination number γtL(G). We show that, for a tree TT of order n≥3n≥3 with ll leaves and ss support vertices, n+l+12−s≤γtL(T)≤n+l2. Moreover, we constructively characterize the extremal trees achieving these bounds.