Hop Italian Domination in Graphs

Abstract

Given a simple graph $G=(V(G),E(G))$, a function $f:V(G)\to \{0,1,2\}$ is a hop Italian dominating function if for every vertex $v$  with $f(v)=0$ there exists a vertex $u$  with $f(u)=2$ for which $u$ and $v$ are of distance $2$ from each other or there exist two vertices $w$ and $z$ for which $f(w)=1=f(z)$ and each of $w$ and $z$ is of distance $2$ from $v$. The minimum weight $\sum_{v\in V(G)}f(v)$ of a hop Italian dominating function is the hop Italian domination number of $G$, and is denoted by $\gamma_{hI}(G)$. In this paper, we initiate the study of the hop Italian domination. In particular, we establish some properties of the the hop Italian dominating function and explore the relationships of the hop Italian domination number with the hop Roman domination number \cite{Rad2,Natarajan} and with the $2$-hop domination number \cite{Canoy}. We study the concept under some binary graph operations. We establish tight bounds and determine exact values for their respective hop  Italian domination numbers.