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## James Oxley

Print publication date: 2011

Print ISBN-13: 9780198566946

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780198566946.001.0001

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# Higher Connectivity

Chapter:
(p.291) 8 Higher Connectivity
Source:
Matroid Theory
Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198566946.003.0009

This chapter addresses the question of whether, for arbitrary n, the property of n-connectedness for graphs can be extended to matroids. It focuses on the case when n = 3. The chapter is organized as follows. Sections 8.1 and 8.2 present Tutte's definition of n-connectedness for matroids and establish some basic properties of the matroid connectivity function and the related local connectivity function. Section 8.3 shows that a matroid is 3-connected if and only if it cannot be decomposed as a direct sum or 2-sum. In addition, it proves a theorem of Cunningham and Edmonds that gives a unique 2-sum decomposition of a connected matroid into 3-connected matroids, circuits, and cocircuits. Section 8.4 discusses some properties of the cycle matroids of wheels and of their relaxations, whirls. Section 8.5 proves Tutte's Linking Theorem, a matroid generalization of Menger's Theorem. Section 8.6 considers the differences between matroid n-connectedness and graph n-connectedness and introduces a matroid generalization of the latter. Section 8.7 proves some extremal connectivity results including Tutte's Triangle Lemma. This lemma is a basic tool in the proof of Tutte's Wheels-and-Whirls Theorem, which is given in Section 8.8.

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