# A notion of minor-based matroid connectivity

## TL;DR

This paper characterizes specific matroids based on their connectivity properties, identifying unique conditions under which matroids maintain minors after element deletion or contraction, and linking connectivity to clonal classes.

## Contribution

It establishes that only $U_{1,2}$ and $M( ext{W}_2)$ are special connected matroids with certain minor-preservation properties, and relates $U_{0,1} igoplus U_{1,1}$-connectivity to trivial clonal classes.

## Key findings

- $U_{1,2}$ is the only connected matroid with minor-preservation under deletion/contraction.
- $U_{1,2}$ and $M( ext{W}_2)$ are the only connected matroids with a specific minor-intersection property.
- $U_{0,1} igoplus U_{1,1}$-connectivity characterizes trivial clonal classes.

## Abstract

For a matroid $N$, a matroid $M$ is $N$-connected if every two elements of $M$ are in an $N$-minor together. Thus a matroid is connected if and only if it is $U_{1,2}$-connected. This paper proves that $U_{1,2}$ is the only connected matroid $N$ such that if $M$ is $N$-connected with $|E(M)| > |E(N)|$, then $M \backslash e$ or $M / e$ is $N$-connected for all elements $e$. Moreover, we show that $U_{1,2}$ and $M(\mathcal{W}_2)$ are the only connected matroids $N$ such that, whenever a matroid has an $N$-minor using $\{e,f\}$ and an $N$-minor using $\{f,g\}$, it also has an $N$-minor using $\{e,g\}$. Finally, we show that $M$ is $U_{0,1} \oplus U_{1,1}$-connected if and only if every clonal class of $M$ is trivial.