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 Seymour's 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. (C) 2002 Elsevier Science (USA).