Geodesic
The Double Roman Domatic Number of a Digraph
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 4, pp. 995-1004.

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A double Roman dominating function on a digraph D with vertex set V(D) is defined in [G. Hao, X. Chen and L. Volkmann, Double Roman domination in digraphs, Bull. Malays. Math. Sci. Soc. (2017).] as a function f : V (D) → 0, 1, 2, 3 having the property that if f(v) = 0, then the vertex v must have at least two in-neighbors assigned 2 under f or one in-neighbor w with f(w) = 3, and if f(v) = 1, then the vertex v must have at least one in-neighbor u with f(u) ≥ 2. A set f_1, f_2, . . ., f_d of distinct double Roman dominating functions on D with the property that ∑_i=1^df_i(v)≤3 for each v ∈ V (D) is called a double Roman dominating family (of functions) on D. The maximum number of functions in a double Roman dominating family on D is the double Roman domatic number of D, denoted by d_dR(D). We initiate the study of the double Roman domatic number, and we present different sharp bounds on d_dR(D). In addition, we determine the double Roman domatic number of some classes of digraphs.
Keywords: digraph, double Roman domination, double Roman domatic number 
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Volkmann, Lutz. The Double Roman Domatic Number of a Digraph. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 4, pp. 995-1004. http://geodesic.mathdoc.fr/item/DMGT_2020_40_4_a3/

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