Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285120200601Total double Roman domination in graphs27391394510.22049/cco.2019.26484.1118ENGuoliangHaoCollege of Science, East China University of Technology, Nanchang, P. R. ChinaLutzVolkmannRWTH Aachen University0000-0003-3496-277XDoost AliMojdehUniversity of Mazandaran0000-0001-9373-3390Journal Article20190507Let $G$ be a simple graph with vertex set $V$. A double Roman dominating function (DRDF) on $G$ is a function $f:Vrightarrow{0,1,2,3}$ satisfying that if $f(v)=0$, then the vertex $v$ must be adjacent to at least two vertices assigned $2$ or one vertex assigned $3$ under $f$, whereas if $f(v)=1$, then the vertex $v$ must be adjacent to at least one vertex assigned $2$ or $3$. The weight of a DRDF $f$ is the sum $sum_{vin V}f(v)$. A total double Roman dominating function (TDRDF) on a graph $G$ with no isolated vertex is a DRDF $f$ on $G$ with the additional property that the subgraph of $G$ induced by the set ${vin V:f(v)ne0}$ has no isolated vertices. The total double Roman domination number $gamma_{tdR}(G)$ is the minimum weight of a TDRDF on $G$. In this paper, we give several relations between the total double Roman domination number of a graph and other domination parameters and we determine the total double Roman domination number of some classes of graphs.http://comb-opt.azaruniv.ac.ir/article_13945_dce686282b94fcb96a05edec316a45ef.pdf