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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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Submodular Functions and Matroid Union

Submodular Functions and Matroid Union

(p.404) 11 Submodular Functions and Matroid Union
Matroid Theory

James Oxley

Oxford University Press

This chapter considers several more matroid constructions. The chapter is organized as follows. Section 11.2 considers several applications of submodular functions, one of which is in the proof of Hall's Marriage Theorem, a result that specifies precisely when a family of sets has a transversal. Section 11.3 discusses the operation of matroid union and presents several of its attractive applications. Section 11.4 considers the problem of sticking two matroids together across a common restriction. Much of the discussion there focuses on a generalization of the operation of parallel connection. The last operation is used in Section 11.5 to define a matroid generalization of the graph operation of Δ-Y exchange.

Keywords:   matroids, submodular functions, Hall's Marriage Theorem, matroid union, parallel connection

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