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A set $D$ of vertices of a graph $G$ is isolating if the set of vertices notin $D$ 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 thecharacterisation of unicyclic and block graphs of order $n$ with isolatingnumber equal to $n/3$. Moreover, we provide a family of general graphsattaining this upper bound on the isolation number.

Graphs with isolation number equal to one third of the order