Geodesic
Signed domination and signed domatic numbers of digraphs
Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 3, pp. 415-427

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Let D be a finite and simple digraph with the vertex set V(D), and let f:V(D) → -1,1 be a two-valued function. If ∑_x ∈ N¯[v]f(x) ≥ 1 for each v ∈ V(D), where N¯[v] consists of v and all vertices of D from which arcs go into v, then f is a signed dominating function on D. The sum f(V(D)) is called the weight w(f) of f. The minimum of weights w(f), taken over all signed dominating functions f on D, is the signed domination number γ_S(D) of D. A set f₁,f₂,...,f_d of signed dominating functions on D with the property that ∑_i = 1^d f_i(x) ≤ 1 for each x ∈ V(D), is called a signed dominating family (of functions) on D. The maximum number of functions in a signed dominating family on D is the signed domatic number of D, denoted by d_S(D). In this work we show that 4-n ≤ γ_S(D) ≤ n for each digraph D of order n ≥ 2, and we characterize the digraphs attending the lower bound as well as the upper bound. Furthermore, we prove that γ_S(D) + d_S(D) ≤ n + 1 for any digraph D of order n, and we characterize the digraphs D with γ_S(D) + d_S(D) = n + 1. Some of our theorems imply well-known results on the signed domination number of graphs.
Keywords: digraph, oriented graph, signed dominating function, signed domination number, signed domatic number 
Volkmann, Lutz. Signed domination and signed domatic numbers of digraphs. Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 3, pp. 415-427. http://geodesic.mathdoc.fr/item/DMGT_2011_31_3_a0/
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