Proof of a conjecture on isolation of graphs dominated by a vertex

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 $N\left[D\right]$ of $D$ in $G$ intersects the vertex sets of the $F$-copies contained by $G$ (equivalently, $G-N\left[D\right]$ contains no $F$-copy). Thus, $\iota (G,K_1)$ is the domination number $\gamma \left(G\right)$ 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, $\gamma \left(F\right)=1$ (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 $\iota (G,F)\le \lfloor \frac{m+1}{k+2}\rfloor$ 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 extreme case where $F$ is a star. The result for the other extreme case where $F$ is a clique had been obtained by Fenech, Kaemawichanurat and the present author. The bound is attainable for any $m\ge 0$ unless $1\le m=k\le 2$. New ideas, including deletion methods and divisibility considerations, are introduced in the proof of the conjecture.