  For a connected $n$-vertex graph $G$ and a set $\mathcal{F}$ of graphs, let
$\iota(G,\mathcal{F})$ denote the size of a smallest set $D$ of vertices of $G$
such that the graph obtained from $G$ by deleting the closed neighbourhood of
$D$ contains no graph in $\mathcal{F}$. Let $\mathcal{E}_k$ denote the set of
connected graphs that have at least $k$ edges. By a result of Caro and
Hansberg, $\iota(G,\mathcal{E}_1) \leq n/3$ if $n \neq 2$ and $G$ is not a
$5$-cycle. The author recently showed that if $G$ is not a triangle and
$\mathcal{C}$ is the set of cycles, then $\iota(G,\mathcal{C}) \leq n/4$. We
improve this result by showing that $\iota(G,\mathcal{E}_3) \leq n/4$ if $G$ is
neither a triangle nor a $7$-cycle. Let $r$ be the number of vertices of $G$
that have only one neighbour. We determine a set $\mathcal{S}$ of six graphs
such that $\iota(G,\mathcal{E}_2) \leq (4n - r)/14$ if $G$ is not a copy of a
member of $\mathcal{S}$. The bounds are sharp.
