# Isolation of cycles

## TL;DR

This paper introduces a new graph parameter measuring the minimum vertex set removal to eliminate all cycles, proves an upper bound for connected graphs, and confirms the bound's sharpness, solving an open problem.

## Contribution

It establishes a tight upper bound of n/4 for the cycle-elimination parameter in connected graphs, addressing a previously open problem.

## Key findings

- Bound of n/4 for connected non-triangle graphs
- The bound is proven to be sharp
- Solves an open problem by Caro and Hansberg

## Abstract

For any graph $G$, let $\iota_{\rm c}(G)$ 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 cycle. We prove that if $G$ is a connected $n$-vertex graph that is not a triangle, then $\iota_{\rm c}(G) \leq n/4$. We also show that the bound is sharp. Consequently, we solve a problem of Caro and Hansberg.