Geodesic
Bounds on the Locating-Total Domination Number in Trees
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 25-34.

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Given a graph G = (V, E) with no isolated vertex, a subset S of V is called a total dominating set of G if every vertex in V has a neighbor in S. A total dominating set S is called a locating-total dominating set if for each pair of distinct vertices u and v in V S, N(u) ∩ S ≠ N(v) ∩ S. The minimum cardinality of a locating-total dominating set of G is the locating-total domination number, denoted by γ_t^L(G). We show that, for a tree T of order n ≥ 3 and diameter d, d+1/2≤γ_t^L(T)≤n−d−1/2, and if T has l leaves, s support vertices and s_1 strong support vertices, then γ_t^L(T)≥max{n+l−s+1/2−s+s_1/4,2(n+1)+3(l−s)−s_1/5}. We also characterize the extremal trees achieving these bounds.
Keywords: tree, total dominating set, locating-total dominating set, locating-total domination number 
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Wang, Kun; Ning, Wenjie; Lu, Mei. Bounds on the Locating-Total Domination Number in Trees. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 25-34. http://geodesic.mathdoc.fr/item/DMGT_2020_40_1_a1/

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