Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 1 1 2016 06 01 Bounds on the restrained Roman domination number of a graph 75 82 13556 10.22049/cco.2016.13556 EN H. Abdollahzadeh Ahangar Babol Noshirvani University of Technology S.R. Mirmehdipour Babol Noshirvani University of Technology Journal Article 2016 08 08 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.  https://comb-opt.azaruniv.ac.ir/article_13556_af7da9ddc41c8343edb4835aaab47c2c.pdf