Isolate restrained domination in graphs

Abstract

A dominating set \(D\) of a graph \(G\) is said to be a restrained dominating set (RDS) of \(G\) if every vertex of \(V-D\) has a neighbor in \(V-D\).  A RDS is said to be an isolate restrained dominating set(IRDS) if \(<D>\) has at least one isolated vertex. The minimum cardinality of a minimal IRDS of $G$ is called the isolate restrained domination number(IRDN), denoted by \(\gamma_{r,0}(G)\). This paper contains basic properties of IRDS and gives the IRDN for the families of graphs such as paths, cycles, complete \(k\)-partite graphs and some other graphs.