Isolate restrained perfect domination in graphs

Abstract

For a graph $G$, a subset $D$ of $V(G)$ is said to be a restrained dominating set(RDS) of $G$ if $D$ is a dominating set and every vertex not in $D$ has a neighbor in $V-D$.  A RDS is said to be an isolate restrained perfect dominating set(IRPDS) if $<D>$ has at least one isolated vertex and $D$ is a perfect dominating set. \\ The minimum cardinality of a minimal IRPDS of $G$ is called the isolate restrained perfect domination number(IRDN), denoted by $\gamma_{r,0,p}(G)$. This paper contains basic properties of IRPDS and gives the IRDPN for the families of graphs such as paths, cycles, complete $k$-partite graphs and some other graphs.