On the intersections of circuits and cocircuits in matroids
type: Item
instance of: scholarly article
title: On the intersections of circuits and cocircuits in matroids
zbMATH Open document ID: 0557.05025
DOI: 10.1007/BF02579220
published in: Combinatorica
publication date: 1984-01-01T00:00:00Z
review text: A 3- or 4-element set is called a triad or a quad, respectively, if it is the intersection of a circuit and a cocircuit of a matroid. \textit{P. D. Seymour} [Combinatorica 1, 387-394 (1981; Zbl 0489.05020)] proved that a matroid has a triad if and only if it is non-binary; and then every pair of elements is contained in a triad. The author characterizes those matroids which have a quad. He also shows that if a matroid has a circuit and a cocircuit meeting in more than 4 elements then it has a quad as well. Finally he proves that if a matroid has a quad and is 3-connected then every pair is in a quad.
reviewed by: András Recski
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 3887713
zbMATH Keywords: circuit of a matroid
zbMATH Keywords: intersection
zbMATH Keywords: circuit
zbMATH Keywords: cocircuit of a matroid
zbMATH Keywords: quad
author: James Oxley
MaRDI profile type: Publication
cites work: \(l\)-matrices and a characterization of binary matroids
cites work: A simple theorem on 3-connectivity
cites work: A Combinatorial Model for Series-Parallel Networks
cites work: The Forbidden Minors of Binary Clutters
cites work: Decomposition of regular matroids
cites work: On minors of non-binary matroids
cites work: Connectivity in Matroids
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description: scientific article; zbMATH DE number 3887713