# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2019: 0.755

SCImago Journal Rank (SJR) 2019: 0.600

Rejection Rate (2018-2019): c. 84%

# Discussiones Mathematicae Graph Theory

Article in press

Authors:

L. Volkmann

Title:

The double Roman domatic number of a digraph

Source:

Discussiones Mathematicae Graph Theory

Received: 2018-03-08, Revised: 2018-06-15, Accepted: 2018-06-15, https://doi.org/10.7151/dmgt.2161

Abstract:

A double Roman dominating function on a digraph $D$ with vertex set $V(D)$ is defined in {[G. Hao, X. Chen and L. Volkmann, Double Roman domination in digraphs, Bull. Malays. Math. Sci. Soc. (2017).]} as a function $f:V(D)\rightarrow\{0,1,2,3\}$ having the property that if $f(v)=0$, then the vertex $v$ must have at least two in-neighbors assigned 2 under $f$ or one in-neighbor $w$ with $f(w)=3$, and if $f(v)=1$, then the vertex $v$ must have at least one in-neighbor $u$ with $f(u)\ge 2$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct double Roman dominating functions on $D$ with the property that $\sum_{i=1}^df_i(v)\le 3$ for each $v\in V(D)$ is called a double Roman dominating family (of functions) on $D$. The maximum number of functions in a double Roman dominating family on $D$ is the double Roman domatic number of $D$, denoted by $d_{dR}(D)$. We initiate the study of the double Roman domatic number, and we present different sharp bounds on $d_{dR}(D)$. In addition, we determine the double Roman domatic number of some classes of digraphs.

Keywords:

digraph, double Roman domination, double Roman domatic number