# Bounds on the restrained Roman domination number of a graph

Document Type : Original paper

Authors

1 Babol Noshirvani University of Technology

2 Babol Noshirvani University of Technology

Abstract

A {em Roman dominating function} on a graph $G$ is a function
$f:V(G)rightarrow {0,1,2}$ satisfying the condition that every
vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex
$v$ for which $f(v) =2$. {color{blue}A {em restrained Roman dominating}
function} $f$ is a {color{blue} Roman dominating function if the vertices with label 0 induce
a subgraph with no isolated vertex.} The weight of a restrained Roman dominating function is
the value $omega(f)=sum_{uin V(G)} f(u)$. The minimum weight of a
restrained Roman dominating function of $G$ is called the { em
restrained Roman domination number} of $G$ and denoted by $gamma_{rR}(G)$.
In this paper we establish some sharp bounds for this parameter.

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