Branch-width and well-quasi-ordering in matroids and graphs

Authors:
James F. Geelen;A. M. H. Gerards;Geoff Whittle
Affiliations:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada;CWI, Postbus 94079, 1090 GB Amsterdam, The Netherlands/ and Department of Mathematics and Computer Science, Eindhoven University of Technology, Postbus 513, 5600 MB Eindhoven, The Netherlands;School of Mathematical and Computing Sciences, Victoria University, P.O. Box 600, Wellington, New Zealand
Venue:
Journal of Combinatorial Theory Series B
Year:
2002

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Abstract

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.