Volkmann, Lutz Double Italian and double Roman domination in digraphs. (English) Zbl 1464.05299 J. Comb. Math. Comb. Comput. 113, 97-108 (2020). In this paper, the concept of double Italian domination is introduced for digraphs. A function \(f: V(D) \rightarrow \{0,1,2,3\}\) is a double Italian dominating function of a digraph \(D\) if each vertex \(u\in V(D)\) with \(f(u)\in \{0,1\}\) has \(\sum_{x\in N^{-}[u]} f(x) \ge 3\), where \(N^{-}[u]\) is the closed in-neighborhood of \(u\). The double Italian domination number \(\gamma_{dI}(D)\) is the minimum weight of a double Italian dominating function of \(D\). This invariant is compared with related previously studied invariants such as the domination number and the Italian domination number. Upper and lower bounds on \(\gamma_{dI}(D)\) in terms of the order, the maximum out-degree, and the minimum in-degree of \(D\) are presented. A new lower bound on the double Roman domination number of a digraph is also proved. Reviewer: Sandi Klavžar (Ljubljana) MSC: 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C20 Directed graphs (digraphs), tournaments Keywords:digraph; double Italian dominating function; double Italian domination number; double Roman domination number PDFBibTeX XMLCite \textit{L. Volkmann}, J. Comb. Math. Comb. Comput. 113, 97--108 (2020; Zbl 1464.05299)