Bounds on the restrained Roman domination number of a graph

Document Type : Original paper


Babol Noshirvani University of Technology


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 $f(v) =2$. A {\em restrained Roman dominating} function $f$ is a 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 $\omega(f)=\sum_{u\in V(G)} f(u)$. 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. 


Main Subjects

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