EDGE ITALIAN DOMINATION OF SOME GRAPH PRODUCTS

An edge Italian dominating function (EIDF) of a graph $G=(V,E)$ is a function $f:E(G)\rightarrow\{0,1,2\}$ such that every edge $x$ with $f(x)=0$ is adjacent to some edge $y$ with $f(y)=2$ or adjacent to at least two edges $z_1,z_2$ with$f(z_1)=f(z_2)=1$. The weight of an edge Italian dominating function is the sum $\sum_{x\in E(G)}f(x)$ and the minimum weight of an edge Italian dominating function of $G$ is called the edge Italian domination number of $G$ and is denoted by $\gamma_I^{\textquotesingle} (G)$. In this paper, we determine the edge Italian domination number of some graph products.