Restrained Italian domination in graphs

For a graph $G=(V(G),E(G))$, an Italian dominating function (ID function) $f:V(G)\rightarrow\{0,1,2\}$ has the property that for every vertex $v\in V(G)$ with $f(v)=0$, either $v$ is adjacent to assigned $2$ under $f$ or least two vertices $1$ $f$. The weight of ID $\sum_{v\in V(G)}f(v)$. domination number minimum taken over all functions $G$. In this paper, we initiate study variant functions. A restrained (RID $G$ which subgraph induced by $\{v\in V(G)\mid f(v)=0\}$ no isolated vertices, and $\gamma_{rI}(G)$ RID We first prove problem computing parameter NP-hard, even when restricted bipartite graphs chordal as well planar maximum degree five. $\gamma_{rI}(T)$ tree $T$ order $n\geq3$ different from double star $S_{2,2}$ can be bounded below $(n+3)/2$. Moreover, extremal trees lower bound are characterized in paper. also give some sharp bounds on general characterizations small large $\gamma_{rI}(G)$.