Geodesic
Bounds on the Signed Roman k-Domination Number of a Digraph
Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 1, pp. 67-79

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Let k be a positive integer. A signed Roman k-dominating function (SRkDF) on a digraph D is a function f : V (D) →{−1, 1, 2 } satisfying the conditions that (i) Σ_ x ∈ N^− [v] f(x) ≥ k for each v ∈ V (D), where N^− [v] is the closed in-neighborhood of v, and (ii) each vertex u for which f(u) = −1 has an in-neighbor v for which f(v) = 2. The weight of an SRkDF f is Σ_ v ∈ V (D) f(v). The signed Roman k-domination number γ_sR^k (D) of a digraph D is the minimum weight of an SRkDF on D. We determine the exact values of the signed Roman k-domination number of some special classes of digraphs and establish some bounds on the signed Roman k-domination number of general digraphs. In particular, for an oriented tree T of order n, we show that γ_sR^2 (T) ≥ (n + 3)//2, and we characterize the oriented trees achieving this lower bound.
Keywords: signed Roman k-dominating function, signed Roman k-domination number, digraph, oriented tree 
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Chen, Xiaodan; Hao, Guoliang; Volkmann, Lutz. Bounds on the Signed Roman k-Domination Number of a Digraph. Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 1, pp. 67-79. http://geodesic.mathdoc.fr/item/DMGT_2019_39_1_a6/