# Graphs without large $K_{2,n}$-minors

## TL;DR

This paper characterizes graphs that lack large $K_{2,n}$-minors and shows that certain highly connected graphs with high minimum degree necessarily contain such minors.

## Contribution

It provides a characterization of graphs without large $K_{2,n}$-minors and proves that large, highly connected graphs with high minimum degree must contain these minors.

## Key findings

- Large 3-connected graphs with minimum degree ≥6 contain $K_{2,n}$-minors.
- 4-connected graphs with a high-degree vertex contain $K_{2,n}$-minors.
- Large 5-connected graphs necessarily have $K_{2,n}$-minors.

## Abstract

The purpose of this paper is to characterize graphs that do not have a large $K_{2,n}$-minor. As corollaries, it is proved that, for any given positive integer $n$, every sufficiently large 3-connected graph with minimum degree at least six, every 4-connected graph with a vertex of sufficiently high degree, and every sufficiently large 5-connected graph must have a $K_{2,n}$-minor.