Geodesic
Bounds on the Locating-Domination Number and Differentiating-Total Domination Number in Trees
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 2, pp. 455-462

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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 γ_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] ∩ 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_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.
Keywords: locating-dominating set, differentiating-total dominating set, tree 
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     author = {Rad, Nader Jafari and Rahbani, Hadi},
     title = {Bounds on the {Locating-Domination} {Number} and {Differentiating-Total} {Domination} {Number} in {Trees}},
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Rad, Nader Jafari; Rahbani, Hadi. Bounds on the Locating-Domination Number and Differentiating-Total Domination Number in Trees. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 2, pp. 455-462. http://geodesic.mathdoc.fr/item/DMGT_2018_38_2_a8/