  A double Roman dominating function of a graph $G$ is a function
$f:V(G)\rightarrow \{0,1,2,3\}$ having the property that for each vertex $v$
with $f(v)=0$, there exists $u\in N(v)$ with $f(u)=3$, or there are $u,w\in
N(v)$ with $f(u)=f(w)=2$, and if $f(v)=1$, then $v$ is adjacent to a vertex
assigned at least $2$ under $f$. The double Roman domination number
$\gamma_{dR}(G)$ is the minimum weight $f(V(G))=\sum_{v\in V(G)}f(v)$ among all
double Roman dominating functions of $G$. An outer independent double Roman
dominating function is a double Roman dominating function $f$ for which the set
of vertices assigned $0$ under $f$ is independent. The outer independent double
Roman domination number $\gamma_{oidR}(G)$ is the minimum weight taken over all
outer independent double Roman dominating functions of $G$.
  In this work, we present some contributions to the study of outer independent
double Roman domination in graphs. Characterizations of the families of all
connected graphs with small outer independent double Roman domination numbers,
and tight lower and upper bounds on this parameter are given. We moreover bound
this parameter for a tree $T$ from below by two times the vertex cover number
of $T$ plus one. We also prove that the decision problem associated with
$\gamma_{oidR}(G)$ is NP-complete even when restricted to planar graphs with
maximum degree at most four. Finally, we give an exact formula for this
parameter concerning the corona graphs.
