A note on nongraphic matroids
type: Item
instance of: scholarly article
title: A note on nongraphic matroids
zbMATH Open document ID: 0551.05030
DOI: 10.1016/0095-8956(84)90062-5
author: T. Asano
author: Takao Nishizeki
author: Paul Seymour
published in: Journal of Combinatorial Theory. Series B
publication date: 1984-01-01T00:00:00Z
review text: The following well-known theorem is due to \textit{W. T. Tutte} [Trans. Am. Math. Soc. 90, 527-552 (1959; Zbl 0084.395)]: A matroid M is graphic if and only if it has no minor isomorphic to \(U^ 2_ 4\), \(F_ 7\), \(F^*_ 7\), \({\mathcal M}^*(K_ 5)\) or \({\mathcal M}^*(K_{3,3})\). Recall that \(U^ 2_ 4\) denotes the four-element matroid in which any 2-set is a base, \(F_ 7\) is the Fano matroid, \(F^*_ 7\) is its dual, \(K_ 5\) and \(K_{3,3}\) are the Kuratowski graphs, and \({\mathcal M}^*(K_ 5)\) or \({\mathcal M}^*(K_{3,3})\) are their bond matroids, respectively. In the paper under review, the authors prove the following variant of the above- mentioned Tutte's result: Let \(\{\) e,f,g\(\}\) be a circuit of a 3- connected nongraphic matroid M. Then M has a minor N isomorphic to one of \(U^ 2_ 4\), \(F_ 7\), \({\mathcal M}^*(K_{3,3})\) such that e,f,g\(\in E(N)\).
reviewed by: J.Libicher
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 3877208
zbMATH Keywords: Fano matroid
zbMATH Keywords: nongraphic matroid
zbMATH Keywords: minor
MaRDI profile type: Publication
full work available at URL: 0095-8956(84)90062-5
OpenAlex ID: W2055484173
cites work: A note on primitive skew curves
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description: scientific article; zbMATH DE number 3877208