Restrained condition on double Roman dominating functions

We continue the study of restrained double Roman domination in graphs. For a graph $G=\big{(}V(G),E(G)\big{)}$, dominating function $f$ is called (RDRD function) if subgraph induced by $\{v\in V(G)\mid f(v)=0\}$ has no isolated vertices. The number number) $\gamma_{rdR}(G)$ minimum weight $\sum_{v\in V(G)}f(v)$ taken over all RDRD functions $G$. first prove that problem computing $\gamma_{rdR}$ NP-hard even for planar graphs, but it solvable linear time when restricted to bounded clique-width graphs such as trees, cographs and distance-hereditary Relationships between some well-known parameters $\gamma_{r}$, $\gamma$ $\gamma_{rR}$ are investigated this paper bounding from below above involving general respectively. $\gamma_{rdR}(T)\geq n+2$ any tree $T\neq K_{1,n-1}$ order $n\geq2$ characterize family trees attaining lower bound. characterization with small numbers given paper.