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 $v_0={vin v(g): f(v)=0}$. 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 $vin v(g)$ 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.