Some extremal connectivity results for matroids
type: Item
instance of: scholarly article
title: Some extremal connectivity results for matroids
zbMATH Open document ID: 0686.05011
DOI: 10.1016/0095-8956(91)90070-Z
author: Safwan Akkari
author: James Oxley
published in: Journal of Combinatorial Theory. Series B
publication date: 1991-01-01T00:00:00Z
review text: \textit{P. D. Seymour} [J. Comb. Theory, Ser. B 26, 159-173 (1979; Zbl 0443.05029)] proved that the 4-point line is the only connected matroid with more than one element in which every 2-element deletion and every 2- element contraction is disconnected, but every 1-element deletion and every 1-element contraction is connected. This paper proves the following result from which one can derive a generalization of Seymour's theorem. Let M be a matroid having at least \(n+2\) elements where n is an integer exceeding one. Then every n-element subset X of E(M) is in an \((n+1)\)- element circuit if and only if (i) for every such subset M/X is disconnected, and (ii) for every smaller subset Y, M/Y is connected. Among the other consequences of this theorem that are obtained is a characterization, in terms of a connectivity condition, of Sylvester matroids, a class of matroids introduced by \textit{U. S. R. Murty} [Aequationes Math. 4, 44-50 (1970; Zbl 0196.027)].
reviewed by: James Oxley
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 4123747
zbMATH Keywords: minimally connected matroids
zbMATH Keywords: n-connected matroid
zbMATH Keywords: Sylvester matroids
MaRDI profile type: Publication
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full work available at URL: 0095-8956(91)90070-z
OpenAlex ID: W2088907071
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description: scientific article; zbMATH DE number 4123747