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 ${\displaystyle {\sum }_{x\in N¯\left[v\right]}f\left(x\right)\ge 1}$ 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 ${\displaystyle {\gamma }_S\left(D\right)}$ of D. A set ${\displaystyle \{f₁,f₂,...,f_d\}}$ of signed dominating functions on D with the property that ${\displaystyle {\sum }_{i=1}^d{f}_i\left(x\right)\le 1}$ 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 ${\displaystyle d_S\left(D\right)}$. In this work we show that ${\displaystyle 4-n\le {\gamma }_S\left(D\right)\le n}$ 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 ${\displaystyle {\gamma }_S\left(D\right)+d_S\left(D\right)\le n+1}$ for any digraph D of order n, and we characterize the digraphs D with ${\displaystyle {\gamma }_S\left(D\right)+d_S\left(D\right)=n+1}$. Some of our theorems imply well-known results on the signed domination number of graphs.