On the total Italian domination number in digraphs

Consider a finite simple digraph $D$ with vertex set $V(D)$. An Italian dominating function (IDF) on $D$ is a function $f:V(D)\to \{0,1,2\}$ satisfying every vertex $u$ with $f(u)=0$ has an in-neighbor $v$ with $f(v)=2$ or two in-neighbors $w$ and $z$ with $f(w)=f(z)=1$. A total Italian dominating function (TIDF) on $D$ is an IDF $f$ such that the subdigraph $D[\{u\mathrm{\mid }f(u)\ge 1\}]$ contains no isolated vertices. The weight $\omega (f)$ of a TIDF $f$ on $D$ is ${\sum }_{u\in V(D)}f(u)$. The total Italian domination number of $D$ is γ_{tI}(D)=\min\{ ω(f)\, |\, \mbox{f$isaTIDFon$D}\}. In this paper, we present bounds on ${\gamma }_{tI}(D)$, and investigate the relationship between several different domination parameters. In particular, we give the total Italian domination number of the Cartesian products $P_2\mathrm{\square }{P}_n$ and $P_3\mathrm{\square }{P}_n$, where $P_n$ represents a dipath with $n$ vertices.