Geodesic
A Note on Roman Domination of Digraphs
Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 1, pp. 13-21

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A vertex subset S of a digraph D is called a dominating set of D if every vertex not in S is adjacent from at least one vertex in S. The domination number of a digraph D, denoted by γ(D), is the minimum cardinality of a dominating set of D. A Roman dominating function (RDF) on a digraph D is a function f : V (D) →0, 1, 2 satisfying the condition that every vertex v with f(v) = 0 has an in-neighbor u with f(u) = 2. The weight of an RDF f is the value ω (f) = Σ_ v ∈ V(D) f(v). The Roman domination number of a digraph D, denoted by γ_R (D), is the minimum weight of an RDF on D. In this paper, for any integer k with 2 ≤ k ≤γ(D), we characterize the digraphs D of order n ≥ 4 with δ − (D) ≥ 1 for which γ_R(D) = (D) + k holds. We also characterize the digraphs D of order n ≥ k with γ_R(D) = k for any positive integer k. In addition, we present a Nordhaus-Gaddum bound on the Roman domination number of digraphs.
Keywords: Roman domination number, domination number, digraph, Nordhaus-Gaddum 
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Chen, Xiaodan; Hao, Guoliang; Xie, Zhihong. A Note on Roman Domination of Digraphs. Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 1, pp. 13-21. http://geodesic.mathdoc.fr/item/DMGT_2019_39_1_a1/