Unique isolated 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 neighbour in $V-D$. An RDS is said to be unique isolated restrained dominating set(UIRDS) if $<D>$ has exactly one isolated vertex. \\ The minimum cardinality of a minimal UIRDS of $G$ is called the Unique isolated restrained domination number(UIRDN), denoted by $\gamma_{r,0}^U(G)$. This paper contains basic properties of UIRDS and gives the UIRDN of paths, cycles, complete $k$-partite graphs and some other graphs.