Ismail Sahul Hamid, S. Balamurugan, A. Navaneethakrishnan

Abstract

A set $S$ of vertices of a graph $G$ such that $\left\langle S\right\rangle$ has an isolated vertex is called an \emph{isolate set} of $G$. The minimum and maximum cardinality of a maximal isolate set are called the \emph{isolate number} $i_0(G)$ and the \emph{upper isolate number} $I_0(G)$ respectively. An isolate set that is also a dominating set (an irredundant set) is an $\emph{isolate dominating set} \ (\emph{an isolate irredundant set})$. The \emph{isolate domination number} $\gamma_0(G)$ and the \emph{upper isolate domination number} $\Gamma_0(G)$ are respectively the minimum and maximum cardinality of a minimal isolate dominating set while the \emph{isolate irredundance number} $ir_0(G)$ and the \emph{upper isolate irredundance number} $IR_0(G)$ are the minimum and maximum cardinality of a maximal isolate irredundant set of $G$. The notion of isolate domination was introduced in \cite{sb} and the remaining were introduced in \cite{isrn}. This paper further extends a study of these parameters.