Double Roman domination and domatic numbers of graphs

Volkmann, Lutz

doi:10.22049/cco.2018.26125.1078

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	Double Roman domination and domatic numbers of graphs
Article 5, Volume 3, Issue 1, Winter and Spring 2018, Page 71-77 PDF (409 K)
Document Type: Original paper
DOI: 10.22049/cco.2018.26125.1078
Author
Lutz Volkmann
RWTH Aachen University
Abstract
A double Roman dominating function on a graph $G$ with vertex set $V(G)$ is defined in cite{bhh} as a function $f:V(G)rightarrow{0,1,2,3}$ 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 $w$ with $f(w)=3$, and if $f(v)=1$, then the vertex $v$ must have at least one neighbor $u$ with $f(u)ge 2$. The weight of a double Roman dominating function $f$ is the sum $sum_{vin V(G)}f(v)$, and the minimum weight of a double Roman dominating function on $G$ is the double Roman domination number $gamma_{dR}(G)$ of $G$. A set ${f_1,f_2,ldots,f_d}$ of distinct double Roman dominating functions on $G$ with the property that $sum_{i=1}^df_i(v)le 3$ for each $vin V(G)$ is called in cite{v} a double Roman dominating family (of functions) on $G$. The maximum number of functions in a double Roman dominating family on $G$ is the double Roman domatic number of $G$. In this note we continue the study the double Roman domination and domatic numbers. In particular, we present a sharp lower bound on $gamma_{dR}(G)$, and we determine the double Roman domination and domatic numbers of some classes of graphs.
Keywords
Domination; Double Roman domination number; Double Roman domatic number
Main Subjects
Graph theory


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