Unique 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(UIRPDS) if $<D>$ has exactly one isolated vertex and $D$ is a perfect dominating set. The minimum cardinality of a minimal UIRPDS of $G$ is called the unique isolate restrained perfect domination number(UIRDN), denoted by $\gamma_{r,0,p}^U(G)$. This paper contains basic properties of UIRPDS and gives the UIRDPN for the families of graphs such as paths, cycles, complete $k$-partite graphs and some other graphs.