A copy of a graph $F$ is called an $F$-copy. For any graph $G$, the $F$-isolation number of $G$, denoted by $ι(G,F)$, 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, $ι(G,K_1)$ is the domination number $γ(G)$ of $G$, and $ι(G,K_2)$ is the vertex-edge domination number of $G$. Settling a conjecture of Zhang and Wu, the present first author proved that if $F$ is a $k$-edge graph, $γ(F) = 1$ (that is, $F$ has a vertex that is adjacent to all the other vertices of $F$), and $G$ is a connected $m$-edge graph, then $ι(G,F) \leq \frac{m+1}{k+2} $ unless $G$ is an $F$-copy or $F$ is a $3$-vertex path and $G$ is a $6$-cycle. The graphs that attain the bound were determined for $0 \leq k \leq 1$ by Fenech, Kaemawichanurat and the present first author, and for $k = 2$ by Zhang and Wu. In this paper, we determine them for $k \geq 3$. Our result was also conjectured by Zhang and Wu.

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