Total double Roman domination in graphs

DOAJ (DOAJ: Directory of Open Access Journals)

Abstract

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.

Authors 3

1. Guoliang Hao

   East China University of Technology

   Affiliation as printed

   East China University of Technology

   College of Science, East China University of Technology, Nanchang, P.R. China

2. Lutz Volkmann Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University

3. Doost Ali Mojdeh

   University of Mazandaran

   Affiliation as printed

   University of Mazandaran

   University of Mazandaran,