Proof of a conjecture on isolation of graphs with a universal vertex

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,K₁) is the domination number $γ(G)$ of G, and ι(G,K₂) is the vertex-edge domination number of G. Settling a conjecture of Zhang and Wu, the 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) ≤ m+1/k+2 unless G is an F-copy or F is a $3$-path and G is a $6$-cycle. We prove another conjecture of Zhang and Wu by determining the graphs that attain the bound.