Geodesic
Strong Equality Between the Roman Domination and Independent Roman Domination Numbers in Trees
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 337-346

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A Roman dominating function (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 is the value f(V (G)) = Σ_u ∈ V (G) f(u). An RDF f in a graph G is independent if no two vertices assigned positive values are adjacent. The Roman domination number γ_R (G) (respectively, the independent Roman domination number i_R(G)) is the minimum weight of an RDF (respectively, independent RDF) on G. We say that γ_R(G) strongly equals i_R(G), denoted by γ_R (G) ≡ i_R(G), if every RDF on G of minimum weight is independent. In this paper we provide a constructive characterization of trees T with γ_R(T) ≡ i_R(T).
Keywords: Roman domination, independent Roman domination, strong equality, trees 
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Chellali, Mustapha; Rad, Nader Jafari. Strong Equality Between the Roman Domination and Independent Roman Domination Numbers in Trees. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 337-346. http://geodesic.mathdoc.fr/item/DMGT_2013_33_2_a7/