Restrained double Roman domatic number

Let $G$ be a graph with vertex set $V(G)$. A double Roman dominating function (DRDF) on a graph $G$ is a function $f:V(G)\longrightarrow\{0,1,2,3\}$ having the property that if $f(v)=0$, then the vertex $v$ must have at least two neighbors assigned 2 under $f$ or one neighbor $w$ with $f(w)=3$, and if $f(v)=1$, then the vertex $v$ mus have at least one neighbor $u$ with $f(u)\ge 2$. If $f$ is a DRDF on $G$, then let $V_0=\{v\in V(G): f(v)=0\}$. A restrained double Roman dominating function is a DRDF $f$ having the property that the subgraph induced by $V_0$ does not have an isolated vertex. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct restrained double Roman dominating functions on $G$ with the property that $\sum_{i=1}^df_i(v)\le 3$ for each $v\in V(G)$ is called a restrained double Roman dominating family (of functions) on $G$. The maximum number of functions in a restrained double Roman dominating family on $G$ is the restrained double Roman domatic number of $G$, denoted by $d_{rdR}(G)$. We initiate the study of the restrained double Roman domatic number, and we present different sharp bounds on $d_{rdR}(G)$. In addition, we determine this parameter for some classes of graphs.