Restrained roman domination in graphs

‎A \textit{Roman dominating function} (RDF) on a graph G = (V,E) is‎ ‎defined to be a function f:V \rightarrow \lbrace ۰,۱,۲\rbrace‎ ‎satisfying the condition that every vertex u for which f(u) = ۰ is‎ ‎adjacent to at least one vertex v for which f(v)=۲‎. ‎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 ۰,۱,۲‎ ‎\rbrace satisfying the condition that every vertex u for which‎ ‎f(u) = ۰ is adjacent to at least one vertex v for which f(v)=۲‎ ‎and at least one vertex w for which f(w) = ۰‎. ‎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 Restrained Roman‎ ‎domination number of G and denoted by \gamma_{rR}(G)‎. ‎In this‎ ‎paper‎, ‎we initiate a study of this parameter‎.

Keywords:

domination , Roman domination , Restrained domination