On the total Italian domination number in digraphs

TL;DR

This paper introduces the concept of total Italian domination in digraphs, establishes bounds for this parameter, and computes it explicitly for Cartesian products of paths, advancing understanding of domination parameters in directed graphs.

Contribution

It defines the total Italian domination number for digraphs, provides bounds, and calculates exact values for specific Cartesian product digraphs, extending domination theory.

Findings

Bounds on the total Italian domination number are established.

Exact values are computed for Cartesian products P2×Pn and P3×Pn.

The relationship between total Italian domination and other parameters is analyzed.

Abstract

Consider a finite simple digraph $D$ with vertex set $V(D)$. An Italian dominating function (IDF) on $D$ is a function $f:V(D)\rightarrow\{0,1,2\}$ 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 $D[\{ u\, |\, f(u)\ge 1\}]$ 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 $\gamma_{tI}(D)=\min\{ \omega(f)\, |\, \mbox{$f$ is a TIDF on $D$}\}$. 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 $P_2\Box P_n$ and $P_3\Box P_n$, where $P_n$ represents a dipath with $n$ vertices.