  A maximal double Roman dominating function (MDRDF) on a graph $G=(V,E)$ is a
function $f:V(G)\rightarrow \{0,1,2,3\}$ such that \textrm{(i) }every vertex
$v$ with $f(v)=0$ is adjacent to least two vertices { assigned $2$ or to at
least one vertex assigned $3,$} \textrm{(ii) }every vertex $v$ with $f(v)=1$ is
adjacent to at least one { vertex assigned $2$ or $3$} and \textrm{(iii) }the
set $\{w\in V|~f(w)=0\}$ is not a dominating set of $G $. The weight of a MDRDF
is the sum of its function values over all vertices, and the maximal double
Roman domination number $\gamma _{dR}^{m}(G) $ is the minimum weight of an
MDRDF on $G$. {In this paper, we initiate the study of maximal double Roman
domination. We first show that the problem of determining }$\gamma
_{dR}^{m}(G)$ {is NP-complete for bipartite, chordal and planar graphs. But it
is solvable in linear time for bounded clique-width graphs including trees,
cographs and distance-hereditary graphs. Moreover, we establish various
relationships relating }$\gamma _{dR}^{m}(G)$ to some domination parameters.
{For the class of trees, we show that for every tree }$T$ {of order }$n\geq 4,$
$\gamma _{dR}^{m}(T)\leq \frac{5}{4}n$ {and we characterize all trees attaining
the bound. Finally, the exact values of }$\gamma _{dR}^{m}(G) $ {are given for
paths and cycles.
