Geodesic
Bounds on the Locating Roman Domination Number in Trees
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 1, pp. 49-62

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A Roman dominating function (or just RDF) on a graph G = (V, E) is a function f : V →{ 0, 1, 2 } satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of an RDF f is the value f(V (G)) = Σ_ u ∈ V (G) f(u). An RDF f can be represented as f = (V_0, V_1, V_2), where V_i = { v ∈ V : f(v) = i } for i = 0, 1, 2. An RDF f = (V_0, V_1, V_2) is called a locating Roman dominating function (or just LRDF) if N(u) ∩ V_2 N(v) ∩ V_2 for any pair u, v of distinct vertices of V_0. The locating Roman domination number γ_R^L (G) is the minimum weight of an LRDF of G. In this paper, we study the locating Roman domination number in trees. We obtain lower and upper bounds for the locating Roman domination number of a tree in terms of its order and the number of leaves and support vertices, and characterize trees achieving equality for the bounds.
Keywords: Roman domination number, locating domination number, locating Roman domination number, tree 
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Jafari Rad, Nader; Rahbani, Hadi. Bounds on the Locating Roman Domination Number in Trees. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 1, pp. 49-62. http://geodesic.mathdoc.fr/item/DMGT_2018_38_1_a3/