# Institute of Mathematics

## A note on the double Roman domination number of graphs

#### Xue-gang Chen

###### Received April, 27, 2018.   Published online September 18, 2019.

Abstract:  For a graph $G=(V,E)$, a double Roman dominating function is a function $f\colon V\rightarrow\{0,1,2,3\}$ having the property that if $f(v)=0$, then the vertex $v$ must have at least two neighbors assigned $2$ under $f$ or one neighbor with $f(w)=3$, and if $f(v)=1$, then the vertex $v$ must have at least one neighbor with $f(w)\geq2$. The weight of a double Roman dominating function $f$ is the sum $f(V)=\sum\nolimits_{v\in V}f(v)$. The minimum weight of a double Roman dominating function on $G$ is called the double Roman domination number of $G$ and is denoted by $\gamma_{\rm dR}(G)$. In this paper, we establish a new upper bound on the double Roman domination number of graphs. We prove that every connected graph $G$ with minimum degree at least two and $G\neq C_5$ satisfies the inequality $\gamma_{\rm dR}(G)\leq\lfloor\frac{13}{11}n\rfloor$. One open question posed by R. A. Beeler et al. has been settled.
Keywords:  double Roman domination number; domination number; minimum degree
Classification MSC:  05C69, 05C35
DOI:  10.21136/CMJ.2019.0212-18

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Affiliations:   Xue-gang Chen, Department of Mathematics, North China Electric Power University, No. 2, Beinong road, Changping District, Beijing 102206, P. R. China, e-mail: gxcxdm@163.com

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