EDGE ITALIAN DOMINATION IN GRAPHS

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 $e$ with $f(e)=0$ is adjacent to some edge $e'$ with $f(e')=2$ or at least two edges $e_1, e_2$ with $f(e_1)=f(e_2)=1.$ The weight of an edge Italian dominating function is $\sum_{e\in E(G)}f(e)$. The edge Italian domination number of a graph $G$ is defined as the minimum weight of an edge Italian dominating function of $G$ and is denoted by $\gamma_I'(G)$. In this paper, we initiate a study on the edge Italian domination in graphs.