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