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Matroid Theory$
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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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The Splitter Theorem

The Splitter Theorem

(p.457) 12 The Splitter Theorem
Matroid Theory

James Oxley

Oxford University Press

This chapter states and proves Seymour's Splitter Theorem. This result, which is a generalization of Tutte's Wheels-and-Whirls Theorem (8.8.4), is a very powerful general tool for deriving matroid structure results. The Wheels-and-Whirls Theorem determines when we can find some element in a 3-connected matroid M to delete or contract in order to preserve 3-connectedness. The Splitter Theorem considers when such an element removal is possible that will not only preserve 3-connectedness but will also maintain the presence of an isomorphic copy of some specified minor of M. The chapter illustrates the power of the Splitter Theorem by noting a variety of consequences of it. It also discusses some extensions and generalizations of the theorem.

Keywords:   matroids, Tutte's Wheels-and-Whirls Theorem, 3-connectedness

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