A decomposition theory for matroids. VI: Almost regular matroids
type: Item
instance of: scholarly article
title: A decomposition theory for matroids. VI: Almost regular matroids
zbMATH Open document ID: 0809.05024
DOI: 10.1016/0095-8956(92)90044-X
published in: Journal of Combinatorial Theory. Series B
publication date: 1993-03-10T00:00:00Z
review text: This paper introduces the class of almost regular matroids. These are the binary matroids \(M\) for which there is a partition (del, con) of \(E(M)\) into non-empty sets such that (i) \(M\backslash e\) is regular if \(e\) is in del; (ii) \(M/e\) is regular if \(e\) is in con; and (iii) every circuit contains an even number of members of con while every cocircuit contains an even number of members of del. The author shows that if \(M\) is such a matroid, then \(M\) is connected, its dual is almost regular, and del is a circuit-hyperplane. The main result of the paper is that, by a sequence of series of parallel extensions and \(\Delta Y\) or \(Y\Delta\) operations, every almost regular matroid can be constructed from either the Fano matroid, \(F_ 7\), or a certain single-element extension of \(R_{10}\).
reviewed by: James Oxley
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 130570
zbMATH Keywords: \(\Delta Y\) operation
zbMATH Keywords: matroid decomposition
zbMATH Keywords: almost regular matroids
zbMATH Keywords: binary matroids
zbMATH Keywords: Fano matroid
author: Klaus Truemper
MaRDI profile type: Publication
cites work: Q5528477
cites work: Decomposition of regular matroids
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cites work: On the delta-wye reduction for planar graphs
cites work: Q3249424
full work available at URL: 0095-8956(92)90044-x
OpenAlex ID: W2068183348
description: scientific article; zbMATH DE number 130570