Unavoidable Connected Matroids Retaining a Specified Minor

Abstract

A sufficiently large connected matroid $M$ contains a big circuit or a big cocircuit. Wu showed that we can ensure that $M$ has a big circuit or a big cocircuit containing any chosen element of $M$. In this paper, we prove that, for a fixed connected matroid $N$, if $M$ is a sufficiently large connected matroid having $N$ as a minor, then, up to duality, either $M$ has a big connected minor in which $N$ is a spanning restriction and the deletion of $E(N)$ is a large connected uniform matroid, or $M$ has, as a minor, the $2$-sum of a big circuit and a connected single-element extension or coextension of $N$. In addition, we find a set of unavoidable minors for the class of graphs that have a cycle and a bond with a big intersection.