Covering Italian domination in graphs

For a graph $G=(V(G),E(G))$, an Italian dominating function (ID function) of $G$ is $f:V(G)\rightarrow \{0,1,2\}$ such that for each vertex $v\in V(G)$ with $f(v)=0$, $f(N(v))\geq2$, is, either there $u \in N(v)$ $f (u) = 2$ or are two vertices $x,y\in $f(x)=f(y)=1$. A covering (CID if $f$ ID and $\{v\in V(G)\mid f(v)\neq0\}$ cover set. The domination number number) $\gamma_{cI}(G)$ the minimum weight taken over all CID functions $G$. In this paper, we study in graphs. We show problem computing parameter NP-hard even when restricted to some well-known families graphs, find bounds on parameter. characterize family graphs which their numbers attain upper bound twice as well claw-free whose lower half orders. also give characterizations small large numbers.