Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 5 1 2020 06 01 Total double Roman domination in graphs 27 39 13945 10.22049/cco.2019.26484.1118 EN Guoliang Hao College of Science, East China University of Technology, Nanchang, P. R. China Lutz Volkmann RWTH Aachen University Doost Ali Mojdeh University of Mazandaran Journal Article 2019 05 07 Let $G$ be a simple graph with vertex set $V$. A double Roman dominating function (DRDF) on $G$ is a function $f:V\rightarrow\{0,1,2,3\}$ 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 $\sum_{v\in V}f(v)$. 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 $\{v\in V:f(v)\ne0\}$ 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. https://comb-opt.azaruniv.ac.ir/article_13945_dce686282b94fcb96a05edec316a45ef.pdf