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

TL;DR

This paper proves a conjecture relating the minimal vertex set needed to isolate all subgraphs isomorphic to a given graph F within a larger connected graph G, extending previous results for special cases.

Contribution

It establishes a general upper bound for the F-isolation number in connected graphs when F has a universal vertex, solving a conjecture posed by Zhang and Wu.

Findings

Bound is tight for all m ≥ 0 except when 1 ≤ m = k ≤ 2.

Introduces new deletion and divisibility techniques in graph theory.

Extends previous results from stars and cliques to broader classes of graphs.

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.