# Improved bounds on the cop number when forbidding a minor

## TL;DR

This paper improves bounds on the cop number for graphs excluding a fixed minor, especially for small or sparse minors, leading to tighter bounds for specific classes like $K_{3,t}$-minor-free graphs.

## Contribution

It provides an improved upper bound on the cop number for minor-free graphs, refining previous results especially for small or subdivided minors.

## Key findings

- Enhanced bounds for cop number in minor-free graphs
- Improved bounds for $K_{3,t}$-minor-free graphs
- Tighter bounds for linklessly embeddable graphs

## Abstract

Andreae (1986) proved that the cop number of connected $H$-minor-free graphs is bounded for every graph $H$. In particular, the cop number is at most $|E(H-h)|$ if $H-h$ contains no isolated vertex, where $h\in V(H)$. The main result of this paper is an improvement on this bound, which is most significant when $H$ is small or sparse, for instance when $H-h$ can be obtained from another graph by multiple edge subdivisions. Some consequences of this result are improvements on the upper bound for the cop number of $K_{3,t}$-minor-free graphs, $K_{2,t}$-minor-free graphs and linklessly embeddable graphs.