Isolation of connected graphs

TL;DR

This paper studies the minimum vertex set size needed to eliminate certain subgraphs in connected graphs, improving existing bounds and providing sharp results for various graph families.

Contribution

It refines bounds on vertex removal sets for eliminating edges, cycles, and specific subgraphs, extending previous results with new sharp bounds.

Findings

Bound of n/3 for vertex sets removing edges, except for a 5-cycle.

Improved bound of n/4 for removing cycles, excluding triangles and 7-cycles.

New bound involving (4n - r)/14 for removing edges, with sharpness proven.

Abstract

For a connected $n$-vertex graph $G$ and a set $\mathcal{F}$ of graphs, let $\iota(G,\mathcal{F})$ denote the size of a smallest set $D$ of vertices of $G$ such that the graph obtained from $G$ by deleting the closed neighbourhood of $D$ contains no graph in $\mathcal{F}$. Let $\mathcal{E}_k$ denote the set of connected graphs that have at least $k$ edges. By a result of Caro and Hansberg, $\iota(G,\mathcal{E}_1) \leq n/3$ if $n \neq 2$ and $G$ is not a $5$-cycle. The author recently showed that if $G$ is not a triangle and $\mathcal{C}$ is the set of cycles, then $\iota(G,\mathcal{C}) \leq n/4$. We improve this result by showing that $\iota(G,\mathcal{E}_3) \leq n/4$ if $G$ is neither a triangle nor a $7$-cycle. Let $r$ be the number of vertices of $G$ that have only one neighbour. We determine a set $\mathcal{S}$ of six graphs such that $\iota(G,\mathcal{E}_2) \leq (4n - r)/14$ if $G$ is not a copy of a member of $\mathcal{S}$. The bounds are sharp.