A.M.H. Gerards

Branch-Width and Well-Quasi-Ordering in Matroids and Graphs

We prove that a class of matroids representable over a fixed finite field and with bounded branch-width is well-quasi-ordered under taking minors. With some extra work, the result implies Robertson and Seymours result that graphs with bounded tree-width (or equivalently, bounded branch-width) are well-quasi-ordered under taking minors. We will not only derive their result from our result on matroids, but we will also use the main tools for a direct proof that graphs with bounded branch-width are well-quasi-ordered under taking minors. This proof also provides a model for the proof of the result on matroids, with all specific matroid technicalities stripped off.

The excluded minors for GF (4)-representable matroids

Abstract There are exactly seven excluded minors for the class of GF (4)-representable matroids.

On the excluded minors for the matroids of branch-width k

We prove that the excluded minors for the class of matroids of branch-width k have size at most (6k - 1)/5.

Disjoint cocircuits in matroids with large rank

We prove that, for any positive integers n, k and q, there exists an integer R such that, if M is a matroid with no M(Kn)- or U2,q+2-minor, then either M has a collection of k disjoint cocircuits or M has rank at most R. Applied to the class of cographic matroids, this result implies the edge-disjoint version of the Erdos-Posa Theorem.