Theory of matroids

337 Pages·2008·2.8 MB·English

by  Neil White

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
THEORY
OF
MATROIDS
Edited by
NEIL WHITE
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
Volume 26
Theory of Matroids
ENCYCLOPEDIA OF MATHEMATICS
and Its Applications
GIAN-CARLO ROTA, Editor
Massachusetts Institute of Technology
Editorial Board
Janos D. Aczel, Waterloo Donald E. Knuth, Stanford
George E. Andrews, Penn State Joshua Lederberg, Rockefeller
Richard Askey, Madison Andrb Lichnerowicz, College de France
Michael F. Atiyah, Oxford M. J. Lighthill, London
Donald Babbitt, U.C.L.A. Chia-Chiao Lin, M.I.T.
Lipman Bers, Columbia Jacques-Louis Lions, Paris
Garrett Birkhoff, Harvard G. G. Lorentz, Austin
Raoul Bott, Harvard Roger Lyndon, Ann Arbor
James K. Brooks, Gainesville Robert J. McEliece, Caltech
Felix E. Browder, Chicago Henry McKean, Courant
A. P. Calderon, Buenos Aires Marvin Marcus, Santa Barbara
Peter A. Carruthers, Los Alamos N. Metropolis, Los Alamos
S. Chandrasekhar, Chicago Frederick Mosteller, Harvard
S. S. Chem, Berkeley Jan Mycielski, Boulder
Hermann Chernoff, M.I.T. L. Nachbin, Rio de Janeiro and Rochester
P. M. Cohn, Bedford College, London Steven A. Orszag, M.I.T.
H. S. MacDonald Coxeter, Toronto Alexander Ostrowski, Basel
George B. Dantzig, Stanford Roger Penrose, Oxford
Nelson Dunford, Sarasota, Florida Carlo Pucci, Florence
F. J. Dyson, Inst. for Advanced Study Fred S. Roberts, Rutgers
Harold M. Edwards, Courant Abdus Salam, Trieste
Harvey Friedman, Ohio State M. P. Schutzenberger, Paris
Giovanni Gallavotti, Rome Jacob T. Schwartz, Courant
Andrew M. Gleason, Harvard Irving Segal, M.I.T.
James Glimm, Courant Oved Shisha, Univ. of Rhode Island
M. Gordon, Essex I. M. Singer, Berkeley
Elias P. Gyftopoulos, M.I.T. Olga Taussky, Caltech
Peter Henrici, ETH, Zurich Ren6 Thom, Bures-sur-Yvette
Nathan Jacobson, Yale John Todd, Caltech
Mark Kac, U.S.C. John W. Tukey, Princeton
Shizuo Kakutani, Yale Veeravalli S. Varadarajan, U.C.L.A.
Samuel Karlin, Stanford Antoni Zygmund, Chicago
J. F. C. Kingman, Oxford
For other books in this series see page 317
THEORY OF MATROIDS
Edited by
Neil White
University of Florida
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CAMBRIDGE UNIVERSITY PRESS
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London New York New Rochelle
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CAMBRIDGE UNIVERSITY PRESS
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Published in the United States of America by Cambridge University Press, New York
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Information on this title: www.cambridge.org/9780521309370
© Cambridge University Press 1986
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 1986
This digitally printed version 2008
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication data
Main entry under title:
Theory of matroids.
(Encyclopedia of mathematics and its applications; v. 26)
Bibliography: p.
1. Matroids. I. White, Neil. II. Series.
QA166.6.T44 1986 511'.6 85-6682
ISBN 978-0-521-30937-0 hardback
ISBN 978-0-521-09202-9 paperback
Contents
List of Contributors page ix
Series Editor's Statement xi
Foreword by Gian-Carlo Rota xiii
Preface xv
Chapter 1 Examples and Basic Concepts 1
Henry Crapo
1.1 Examples from Linear Algebra and Projective
Geometry
1
1.2 Further Algebraic Examples 12
1.3 Combinatorial Examples 16
1.4 Structure and Related Geometries 25
Exercises 27
Chapter 2 Axiom Systems 29
Giorgio Nicoletti and Neil White
2.1 Basis Axioms 30
2.2 Other Families of Subsets 32
2.3 Closure and Rank 38
2.4 Combinatorial Geometries and Infinite Matroids 42
V
Contents
Exercises 43
References 44
Chapter 3 Lattices 45
Ulrich Faigle
3.1 Posets and Lattices 46
3.2 Modularity 49
3.3 Semimodular Lattices of Finite Length 51
3.4 Geometric Lattices 53
3.5 Decomposition of Geometric Lattices 56
3.6 Projective Geometry and Modular Geometric
Lattices 58
Exercises 60
References 61
Chapter 4 Basis-Exchange Properties 62
Joseph P. S. Kung
4.1 Bracket Identities and Basis-Exchange Properties 62
4.2 The Exchange Graph 64
4.3 Multiple and Alternating Exchanges 66
Historical Notes 69
Exercises 69
References 73
Chapter 5 Orthogonality 76
Henry Crapo
5.1 Introduction 76
5.2 Orthogonal Geometries 77
5.3 Vector Geometries and Function-Space Geometries 81
5.4 Orthogonality of Vector Geometries 85
5.5 Orthogonality of Simplicial Geometries 87
5.6 Orthogonality of Planar Graphic Geometries 90
5.7 Research Problem: Orthogonality between
Other Pairs of Simplicial Geometries 91
5.8 The Orthogonal of a Structure Geometry 94
References 96
Chapter 6 Graphs and Series-Parallel Networks 97
James Oxley
6.1 Polygon Matroids, Bond Matroids, and Planar
Graphs 98
6.2 Connectivity for Graphs and Matroids 107
6.3 Whitney's 2-Isomorphism Theorem 110
6.4 Series-Parallel Networks 116
Contents Vii
Exercises 120
References 125
Chapter 7 Constructions 127
Thomas Brylawski
7.1 Introduction 127
7.2 Isthmuses and Loops 128
7.3 Deletions, Submatroids, and Extensions 130
7.4 Contractions, Minors, and Lifts 138
7.5 Truncations, Lifts, and Matroid Bracing 162
7.6 Direct Sum and Its Generalizations 173
7.7 Lower Truncations 193
7.8 Index of Constructions 201
Exercises 209
References 222
Chapter 8 Strong Maps 224
Joseph P. S. Kung
8.1 Minors and Strong Maps 224
8.2 The Factorization Theorem 230
8.3 Elementary Quotient Maps 237
8.4 Further Topics 240
Historical Notes 242
Exercises 243
References 252
Chapter 9 Weak Maps 254
Joseph P. S. Kung and Hien Q. Nguyen
9.1 The Weak Order 254
9.2 Weak Cuts 256
9.3 Rank-preserving Weak Maps 260
9.4 Simple Weak Maps of Binary Matroids 262
Historical Notes 267
Exercises 268
References 270
Chapter 10 Semimodular Functions 272
Hien Q. Nguyen
10.1 General Properties of Semimodular Functions 273
10.2 Expansions and Dilworth's Embedding 275
10.3 Reductions 282
10.4 Applications of Expansions and Reductions 289
Historical Notes 296
References 297
viii Contents
Appendix of Matroid Cryptomorphisms 298
Thomas Brylawski
Axiomatizations for the Matroid M(E) 300
Cryptomorphisms 304
Prototypical Examples 305
Special Cryptomorphisms Characterizing Binary
Matroids 310
Index 313
Contributors
Thomas Brylawski Department of Mathematics
University of North Carolina
Chapel Hill, NC 27514
Henry Crapo Institut National de Recherche en
Informatique et en Automatique, B. P. 105
78153 le Chesnay Cedex, France
Ulrich Faigle Institut fur Okonometrie and
Operations Research, Universitat Bonn
Nassestrasse 2, D-5300 Bonn 1
Federal Republic of Germany
Joseph P. S. Kung Department of Mathematics
North Texas State University
Denton, TX 76203
Hien Q. Nguyen Department of Mathematics
University of Montana
Missoula, MT 59812
Giorgio Nicoletti Istituto di Geometria
Universita Bologna
Piazza di Porta S. Donato
Bologna, Italy
ix