  A copy of a graph $F$ is called an $F$-copy. For any graph $G$, the
$F$-isolation number of $G$, denoted by $\iota(G,F)$, is the size of a smallest
subset $D$ of the vertex set of $G$ such that the closed neighbourhood $N[D]$
of $D$ in $G$ intersects the vertex sets of the $F$-copies contained by $G$
(equivalently, $G-N[D]$ contains no $F$-copy). Thus, $\iota(G,K_1)$ is the
domination number $\gamma(G)$ of $G$, and $\iota(G,K_2)$ is the vertex-edge
domination number of $G$. We prove that if $F$ is a $k$-edge graph, $\gamma(F)
= 1$ (that is, a vertex of $F$ is adjacent to all the other vertices of $F$),
and $G$ is a connected $m$-edge graph, then $\iota(G,F) \leq \big\lfloor
\frac{m+1}{k+2} \big\rfloor$ unless $G$ is an $F$-copy or $F$ is a $3$-path and
$G$ is a $6$-cycle. This was recently posed as a conjecture by Zhang and Wu,
who settled the case where $F$ is a star. The result for the case where $F$ is
a clique had been obtained by Fenech, Kaemawichanurat and the present author.
The bound is attainable for any $m \geq 0$ unless $m = k \leq 2$. New ideas,
such as the consideration of divisibility, are introduced in the proof of the
conjecture.
