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 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. We prove that if F is a k-edge graph, γ(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 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. 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 ≥ 0 unless m = k ≤ 2. New ideas, such as the consideration of divisibility, are introduced in the proof of the conjecture.