Graphs with isolation number equal to one third of the order

TL;DR

This paper characterizes unicyclic and block graphs with the maximum possible isolation number of one third of their order, and introduces a family of graphs that reach this upper bound.

Contribution

It provides a complete characterization of certain graph classes with maximum isolation number and constructs a family of graphs achieving this bound.

Findings

Unicyclic and block graphs with isolation number n/3 are characterized.

A family of graphs attaining the upper bound on isolation number is constructed.

The maximum isolation number for connected graphs of order n (except C5) is n/3.

Abstract

A set $D$ of vertices of a graph $G$ is isolating if the set of vertices not in $D$ or with no neighbor in $D$ is independent. The isolation number of $G$, denoted by $\iota (G)$, is the minimum cardinality of an isolating set of $G$. It is known that $\iota (G)\le n/3$, if $G$ is a connected graph of order $n$, $n\ge 3$, distinct from $C_5$. The main result of this work is the characterisation of unicyclic and block graphs of order $n$ with isolating number equal to $n/3$. Moreover, we provide a family of general graphs attaining this upper bound on the isolation number.