A chain theorem for 4-connected matroids
type: Item
instance of: scholarly article
title: A chain theorem for 4-connected matroids
zbMATH Open document ID: 1058.05015
author: Rhiannon Hall
published in: Journal of Combinatorial Theory. Series B
publication date: 2005-02-09T00:00:00Z
review text: Much attention has been given to the study of 3-connected matroids. Inductive arguments can be made in this class of matroids via the use of Tutte's wheels and whirls theorem and Seymour's splitter theorem. Matroid representation theory requires a higher connectivity and strict 4-connectivity is too strong. Studying weakenings of 4-connectivity is important in furthering the study of matroid representations. In the paper under review, the author focusses on weakening 4-connectivity by focusing on the size of 3-connectors as opposed the structure of 3-connectors. The main theorem of the paper proves that if a 3-connected matroid \(M\) is 4-connected up to 3-separators of size 5, then there is an element \(x\in E(M)\) such that either co\((M\setminus x)\) or si\((M/x)\) is 3-connected and 4-connected up to 3-separators of size 5, and has a cardinality of \(| E(M)| -1\) or \(| E(M)| -2\). In addition, a theorem useful in bounding the size of excluded minors for matroids of branch-width 3 is proven.
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 2132707
zbMATH Keywords: 3-connected matroids
zbMATH Keywords: Chain theorem
zbMATH Keywords: 3-separator
reviewed by: Kelly J. Pearson
MaRDI profile type: Publication
full work available at URL: j.jctb.2004.06.003
OpenAlex ID: W2048456408
cites work: Branch-width and well-quasi-ordering in matroids and graphs.
cites work: Matroid 4-connectivity: A deletion-contraction theorem
cites work: Q3135082
cites work: The structure of a 3-connected matroid with a 3-separating set of essential elements
cites work: Connectivity in Matroids
DOI: 10.1016/J.JCTB.2004.06.003
description: scientific article; zbMATH DE number 2132707