Geodesic
The Distance Roman Domination Numbers of Graphs
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 4, pp. 717-730 Cet article a éte moissonné depuis la source Library of Science

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Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-distance Roman dominating function on G is a labeling f : V (G) → 0, 1, 2 such that for every vertex with label 0, there is a vertex with label 2 at distance at most k from each other. The weight of a k-distance Roman dominating function f is the value ω (f) =∑_v∈V f(v). The k-distance Roman domination number of a graph G, denoted by γ_R^k (D), equals the minimum weight of a k-distance Roman dominating function on G. Note that the 1-distance Roman domination number γ_R^1 (G) is the usual Roman domination number γ_R (G). In this paper, we investigate properties of the k-distance Roman domination number. In particular, we prove that for any connected graph G of order n ≥ k +2, γ_R^k (G) ≤ 4n//(2k +3) and we characterize all graphs that achieve this bound. Some of our results extend these ones given by Cockayne et al. in 2004 and Chambers et al. in 2009 for the Roman domination number.
Keywords: k-distance Roman dominating function, k-distance Roman domination number, Roman dominating function, Roman domination number 
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Aram, Hamideh; Norouzian, Sepideh; Sheikholeslami, Seyed Mahmoud. The Distance Roman Domination Numbers of Graphs. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 4, pp. 717-730. http://geodesic.mathdoc.fr/item/DMGT_2013_33_4_a7/

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