H. Abdollahzadeh Ahangar, M. Chellali, S.M. Sheikholeslami, J.C. Valenzuela-Tripodoro

A maximal double Roman dominating function (MDRDF) on a graph $G=(V,E)$ is a function $f:V\left(G\right)\to \{0,1,2,3\}$ such that (i) every vertex $v$ with $f\left(v\right)=0$ is adjacent to least two vertices assigned 2 or to at least one vertex assigned 3, (ii) every vertex $v$ with $f\left(v\right)=1$ is adjacent to at least one vertex assigned 2 or 3 and (iii) the set $\{w\in V|\phantom{\rule{0.33em}{0ex}}f\left(w\right)=0\}$ 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 ${\gamma}_{dR}^{m}\left(G\right)$ 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 ${\gamma}_{dR}^{m}\left(G\right)$ 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 ${\gamma}_{dR}^{m}\left(G\right)$ to some domination parameters. For the class of trees, we show that for every tree $T$ of order $n\ge 4,$ ${\gamma}_{dR}^{m}\left(T\right)\le \frac{5}{4}n$ and we characterize all trees attaining the bound. Finally, the exact values of ${\gamma}_{dR}^{m}\left(G\right)$ are given for paths and cycles.