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 $\sum _{x\in N^{-}[u]}f(x)\ge 3,$ 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.