Italian Domination in Digraphs

Abstract

An \emph{Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function $f:V(D)\to \{0,1,2\}$ such that every vertex $v\in V(D)$ with $f(v)=0$ has at least two in-neighbors assigned 1 under $f$ or one in-neighbor $w$ with $f(w)=2$. The weight of an Italian dominating function is the sum $\sum _{v\in V(D)}f(v)$, and the minimum weight of an Italian dominating function $f$ is the \emph{Italian domination number}, denoted by ${\gamma }_I(D)$. We initiate the study of the Italian domination number for digraphs, and we present different sharp bounds on ${\gamma }_I(D)$. In addition, we determine the Italian domination number of some classes of digraphs. As applications of the bounds and properties on the Italian domination number in digraphs, we give some new and some known results of the Italian domination number in graphs.