On chains of 3-connected matroids
type: Item
instance of: scholarly article
title: On chains of 3-connected matroids
zbMATH Open document ID: 0609.05024
DOI: 10.1016/0166-218X(86)90039-9
author: Robert E. Bixby
author: Collette R. Coullard
published in: Discrete Applied Mathematics
publication date: 1986-01-01T00:00:00Z
review text: A sequence \(N_ 0,N_ 1,...,N_ m\) of 3-connected matroids is called a chain from \(N_ 0\) to \(N_ m\) if each of \(N_ 0,N_ 1,...,N_{m-1}\) is a minor of its successor. The gap of such a chain is \(\max \{| E(N_ i)| -| E(N_{i-1})|:\) \(i=1,2,...,m\}\). A result of \textit{K. Truemper} [Eur. J. Comb. 5, 377-394 (1984; Zbl 0561.05018)] shows that if N is a 3-connected minor of a 3-connected matroid M, then there is always a chain from N to M of gap at most three and this paper proves several extensions of Truemper's result. In particular, the authors show that the chain of gap at most three from N to M may be chosen so that each term is obtained from its successor by a single-element contraction followed by a simplification, or by a single-element deletion followed by a cosimplification. In addition, they prove that, provided N has no circuits or cocircuits of size less than four, one may find a chain whose gap is at most two.
reviewed by: James Oxley
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 3985241
zbMATH Keywords: 3-connected matroids
zbMATH Keywords: chain
zbMATH Keywords: 3-connected minor
MaRDI profile type: Publication
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full work available at URL: 0166-218x(86)90039-9
OpenAlex ID: W1972502774
description: scientific article; zbMATH DE number 3985241