Covering total double Roman domination in graphs

For a graph $G$ with no isolated vertex, a covering total double Roman dominating function ($CTDRD$ function) $f$ of $G$ is a total double Roman dominating function ($TDRD$ function) of $G$ for which the set ${vin V(G)| f(v)ne 0}$ is a vertex cover set. The covering total double Roman domination number $gamma_{ctdR}(G)$ equals the minimum weight of an $CTDRD$ function on $G$. An $CTDRD$ function on $G$ with weight $gamma_{ctdR} (G)$ is called a $gamma_{ctdR} (G)$-function. In this paper, the graphs $G$ with small $gamma_{ctdR} (G)$ are characterised. We show that the decision problem associated with $CTDRD$ is $NP$-complete even when restricted to planer graphs with maximum degree at most four. We then show that for every graph $G$ without isolated vertices, $gamma_{oitR}(G)<gamma_{ctdR}(G)< 2gamma_{oitR}(G)$ and for every tree $T$, $2beta(T)+1leq gamma_{ctdR}(T)leq4beta(T)$, where $gamma_{oitR}(G)$ and $beta(T)$ are the outer independent total Roman domination number of $G$, and the minimum vertex cover number of $T$ respectively. Moreover we investigate the $gamma_{ctdR}$ of corona of two graphs.