A double Roman dominating function of a graph $G$ is $f:V(G)\rightarrow \{0,1,2,3\}$ having the property that for each vertex $v$ with $f(v)=0$, there exists $u\in N(v)$ $f(u)=3$, or are $u,w\in $f(u)=f(w)=2$, and if $f(v)=1$, then adjacent to assigned at least $2$ under $f$. The domination number $\gamma_{dR}(G)$ minimum weight $f(V(G))=\sum_{v\in V(G)}f(v)$ among all functions $G$. An outer i...
