Remarks on the restrained Italian domination number in graphs

Let $G$ be a graph with vertex set $V(G)$. An Italian dominating function (IDF) is a function $f:V(G)longrightarrow {0,1,2}$ having the property that that $f(N(u))geq 2$ for every vertex $uin V(G)$ with $f(u)=0$, where $N(u)$ is the neighborhood of $u$. If $f$ is an IDF on $G$, then let $V_0={vin V(G): f(v)=0}$. A restrained Italian dominating function (RIDF) is an Italian dominating function $f$ having the property that the subgraph induced by $V_0$ does not have an isolated vertex. The weight of an RIDF $f$ is the sum $sum_{vin V(G)}f(v)$, and the minimum weight of an RIDF on a graph $G$ is the restrained Italian domination number. We present sharp bounds for the restrained Italian domination number, and we determine the restrained Italian domination number for some families of graphs.

* The formulas are not displyed corrrectly.