Restrained roman domination in graphs

Abstract

‎A textit{Roman dominating function} (RDF) on a graph $G = (V,E)$ is‎ ‎defined to be a function $ f:V rightarrow lbrace 0,1,2rbrace$‎ ‎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 set $S‎ ‎subseteq V$ is a textit{Restrained dominating set} if every vertex‎ ‎not in $S$ is adjacent to a vertex in $S$ and to a vertex in $V‎ - ‎S$‎. ‎We define a textit{Restrained Roman dominating function} on a‎ ‎graph $G = (V,E)$ to be a function $f‎ : ‎V rightarrow lbrace 0,1,2‎ ‎rbrace$ satisfying the condition that every vertex $u$ for which‎ ‎$f(u) = 0 $ is adjacent to at least one vertex $v$ for which $f(v)=2$‎ ‎and at least one vertex $w$ for which $f(w) = 0$‎. ‎The textit{weight}‎ ‎of a Restrained Roman dominating function is the value $f(V)= sum _{u‎ ‎in V} f(u)$‎. ‎The minimum weight of a Restrained Roman dominating‎ ‎function on a graph $G$ is called the <em>Restrained Roman‎ ‎domination number</em> of $G$ and denoted by $gamma_{rR}(G)$‎. ‎In this‎ ‎paper‎, ‎we initiate a study of this parameter‎.