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

Abstract

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, $F$ has a vertex that is adjacent to all the other vertices of $F$), and $G$ is a connected $m$-edge graph, then $\iota(G,F) \leq \big\lfloor \frac{m+1}{k+2} \big\rfloor$ 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 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 $m \geq 0$ unless $1 \leq m = k \leq 2$. New ideas, including deletion methods and divisibility considerations, are introduced in the proof of the conjecture.