Signed domination and signed domatic numbers of digraphs

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 γS(D) of D. A set {f1, f2, . . . , fd} of signed dominating functions on D with the property that ∑d i=1 fi(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 dS(D). In this work we show that 4 − n ≤ γS(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 γS(D)+ dS(D) ≤ n+ 1 for any digraph D of order n, and we characterize the digraphs D with γS(D) + dS(D) = n+1. Some of our theorems imply well-known results on the signed domination number of graphs.