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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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Minors

Minors

Chapter:
3 Minors
Source:
Matroid Theory
Author(s):

James Oxley

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198566946.003.0004

This chapter begins by introducing the operation of contraction as the dual of the operation of deletion. It then derives a definition of contraction that does not use duality and looks at how contraction affects various special sets such as independent sets, bases, and circuits. It considers some examples of minor-closed classes and shows that the classes of transversal matroids and strict gammoids are not minor-closed. It proves the Scum Theorem, which asserts that the sets X and Y can be chosen so that M/Y has the same rank as N. It also investigate how to obtain geometric representations for M\e and M/e from a geometric representation for M; and examines the effects of the operations of deletion and contraction on the flats of a matroid.

Keywords:   matroids, dual, Scum Theorem, contraction, deletion

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