Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21281120160601Bounds on the restrained Roman domination number of a graph75821355610.22049/cco.2016.13556ENH.Abdollahzadeh AhangarBabol Noshirvani University of Technology0000-0002-0038-8047S.R.MirmehdipourBabol Noshirvani University of TechnologyJournal Article20160808A {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 $omega(f)=sum_{uin V(G)} f(u)$. 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. http://comb-opt.azaruniv.ac.ir/article_13556_af7da9ddc41c8343edb4835aaab47c2c.pdf