# Double Roman domination and domatic numbers of graphs

Document Type: Original paper

Author

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.

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