Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21288120230301Remarks on the restrained Italian domination number in graphs1831911432910.22049/cco.2021.27471.1269ENLutzVolkmannRWTH Aachen University0000-0003-3496-277XJournal Article20211021Let $G$ 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 $u\in V(G)$ with $f(u)=0$, where $N(u)$ is the neighborhood of $u$. If $f$ is an IDF on $G$, then let $V_0=\{v\in V(G): f(v)=0\}$. 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 $\sum_{v\in V(G)}f(v)$, 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.http://comb-opt.azaruniv.ac.ir/article_14329_1ab676fb95dc17392b17649fcfc8bc0a.pdf