Branch-width and well-quasi-ordering in matroids and graphs.
type: Item
instance of: scholarly article
title: Branch-width and well-quasi-ordering in matroids and graphs.
zbMATH Open document ID: 1037.05013
author: A. M. H. Gerards
author: James F. Geelen
author: Geoff Whittle
published in: Journal of Combinatorial Theory. Series B
publication date: 2002-12-10T00:00:00Z
full work available at URL: https://ir.cwi.nl/pub/1100
review text: The authors prove that given a finite field \(F\) and an integer \(n\), each infinite collection of \(F\)-representable matroids with branch-width at most \(n\) has two members such that one is a minor of the other. They show that finiteness of \(F\) is crucial by showing that for each infinite field there exist infinite antichains of branch-width 3 that are representable over that field. The proof ideas are introduced by providing a direct proof of the \textit{N. Robertson} and \textit{P. D. Seymour} analog for graphs in [Graph minors. IV: Tree-width and well-quasi-ordering, J. Comb. Theory, Ser. B 48, No. 2, 227--254 (1990; Zbl 0719.05032)].
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 05C99
zbMATH DE Number: 1843815
zbMATH Keywords: representable matroid
zbMATH Keywords: tree-width
zbMATH Keywords: minors
zbMATH Keywords: finite fields
zbMATH Keywords: submodularity
zbMATH Keywords: connectivity
reviewed by: Herman J. Servatius
MaRDI profile type: Publication
OpenAlex ID: W2006587062
cites work: Highly connected sets and the excluded grid theorem
cites work: Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture
cites work: Q5734436
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cites work: Menger's theorem for matroids
DOI: 10.1006/JCTB.2001.2082
description: scientific article; zbMATH DE number 1843815