Double Italian and double Roman domination in digraphs

Abstract

A double Italian dominating function on a digraph \( D \) with vertex set \( V(D) \) is defined as a function \( f: V(D) \to \{0,1,2,3\} \) such that each vertex \( u \in V(D) \) with \( f(u) \in \{0,1\} \) has the property that \(\sum_{x \in N^{-}[u]} f(x) \geq 3,\) where \( N^{-}[u] \) is the closed in-neighborhood of \( u \). The weight of a double Italian dominating function is the sum \(\sum_{v \in V(D)} f(v),\) and the minimum weight of a double Italian dominating function \( f \) is the double Italian domination number, denoted by \( \gamma_{dI}(D) \). We initiate the study of the double Italian domination number for digraphs, and we present different sharp bounds on \( \gamma_{dI}(D) \). In addition, several relations between the double Italian domination number and other domination parameters such as double Roman domination number, Italian domination number, and domination number are established.