CONTEMPORARY 197

Matroid Theory

AMS-IMS-SIAM Joint Summer Research Conference on Theory July 2-6, 1995 University of Washington, Seattle

Joseph E. Bonin James G. Oxley Brigitte Servatius Editors http://dx.doi.org/10.1090/conm/197

Selected Titles in This Series

197 J. E. Bonin, J. G. Oxley, and B. Servatius, Editors, Matroid theory, 1996 196 David Bao, Shiing-shen Chern, and Zhongmin Shen, Editors, Finsler , 1996 195 Warren Dicks and Enric Ventura, The group fixed by a family of injective endomorphisms of a free group, 1996 194 Seok-Jin Kang, Myung-Hwan Kim, and Insok Lee, Editors, Lie and their representations, 1996 193 Chongying Dong and Geoffrey Mason, Editors, Moonshine, the Monster, and related topics, 1996 192 Tomek Bartoszynski and Marion Scheepers, Editors, Set theory, 1995 191 Tuong Ton-That, Kenneth I. Gross, Donald St. P. llichards, and Paul J. Sally, Jr., Editors, Representation theory and harmonic analysis, 1995 190 Mourad E. H. Ismail, M. Zuhair Nashed, Ahmed I. Zayed, and Ahmed F. Ghaleb, Editors, Mathematical analysis, wavelets, and signal processing, 1995 189 S. A.M. Marcantognini, G. A. Mendoza, M.D. Moran, A. Octavio, and W. 0. Urbina, Editors, Harmonic analysis and operator theory, 1995 188 Alejandro Adem, R. James Milgram, and Douglas C. Ravenel, Editors, Homotopy theory and its applications, 1995 187 G. W. Brumfiel and H. M. Hilden, SL(2) representations of finitely presented groups, 1995 186 Shreeram S. Abhyankar, Walter Feit, Michael D. Fried, Yasutaka Ihara, and Helmut Voelklein, Editors, Recent developments in the inverse Galois problem, 1995 185 Raw E. Curto, Ronald G. Douglas, Joel D. Pincus, and Norberto Salinas, Editors, Multivariable operator theory, 1995 184 L. A. Bokut', A. I. Kostrikin, and S. S. Kutateladze, Editors, Second International Conference on , 1995 183 William C. Connett, Marc-Olivier Gebuhrer, and Alan L. Schwartz, Editors, Applications of hypergroups and related measure algebras, 1995 182 Selman Akbulut, Editor, Real algebraic geometry and , 1995 181 Mila Cenkl and Haynes Miller, Editors, The Cech Centennial, 1995 180 David E. Keyes and Jinchao Xu, Editors, Domain decomposition methods in scientific and engineering computing, 1994 179 Yoshiaki Maeda, Hideki Omoro, and Alan Weinstein, Editors, Symplectic geometry and quantization, 1994 178 Hc!lime Barcelo and Gil Kalai, Editors, Jerusalem '93, 1994 177 Simon Gindikin, Roe Goodman, Frederick P. Greenleaf, and Paul J. Sally, Jr., Editors, Representation theory and analysis on homogeneous spaces, 1994 176 David Ballard, Foundational aspects of "non"standard mathematics, 1994 175 Paul J. Sally, Jr., Moshe Flato, James Lepowsky, Nicolai Reshetikhin, and Gregg J. Zuckerman, Editors, Mathematical aspects of conformal and topological theories and quantum groups, 1994 174 Nancy Childress and John W. Jones, Editors, Arithmetic geometry, 1994 173 Robert Brooks, Carolyn Gordon, and Peter Perry, Editors, Geometry of the spectrum, 1994 172 Peter E. Kloeden and Kenneth J. Palmer, Editors, Chaotic numerics, 1994 171 Riidiger Gobel, Paul Hill, and Wolfgang Liebert, Editors, Abelian group theory and related topics, 1994 170 John K. Deem and Krishan L. Duggal, Editors, Differential geometry and mathematical physics, 1994 (Continued in the back of this publication) Conference Participants CoNTEMPORARY MATHEMATICS

197

Matroid Theory

AMS-IMS-SIAM Joint Summer Research Conference on Matroid Theory July 2-6, 1995 University of Washington, Seattle

Joseph E. Bonin James G. Oxley Brigitte Servatius Editors

American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dennis DeTurck, managing editor Andy Magid Michael Vogelius Clark Robinson Peter M. Winkler

The 1995 AMS-IMS-SIAM Joint Summer Research Conference on Matroid Theory was held at the University of Washington, Seattle, from July 2 to July 6, 1995, with support from National Science Foundation Grant No. DMS-9221892-003 and National Security Agency Grant No. MDA 904-95-1-1063. 1991 Mathematics Subject Classification. Primary 05835, 52C25; Secondary 51E20, 05C15, 68U07.

Library of Congress Cataloging-in-Publication Data AMS-IMS-SIAM Joint Summer Research Conference on Matroid Theory (1995 : University of Washington) Matroid theory : AMS-IMS-SIAM Joint Summer Research Conference on Matroid Theory, July 2-6, 1995, University of Washington, Seattle/ Joseph E. Bonin, James G. Oxley, Brigitte Servatius, editors. p. em. - (Contemporary mathematics, ISSN 0271-4132; 197) Includes bibliographical references (p. - ). ISBN 0-8218-0508-8 (alk. paper) 1. -Congresses. I. Bonin, Joseph E. (Joseph Edmond), 1962- . II. Oxley, J. G. III. Servatius, Brigitte, 1954- . IV. Title. V. Series: Contemporary mathematics (American Mathematical Society); v. 197. QA166.6.A45 1995 511'.6---dc20 96-18251 CIP

Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionOams.org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © 1996 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. § The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. 10 9 8 7 6 5 4 3 2 1 01 00 99 98 97 96 Contents

Preface vii Conference program ix List of participants xi Critical problems JOSEPH p. S. KUNG 1 Structure theory and connectivity for matroids JAMES OXLEY 129 Some matroids from discrete applied geometry WALTER WHITELEY 171 pairs, theory and an electric application SETH CHAIKEN 313 A min-max theorem using matroid separations JACK S. DHARMATILAKE 333 A characteristic polynomial GARY GORDON AND ELIZABETH McMAHON 343 Monotactic matroids ROBERT E. JAMISON 353

On binary matroids with a K 3 ,3-minor S. R. KINGAN 363 Randomised approximation of the number of bases LAURA CHAVEZ LOMELI AND DOMINIC WELSH 371

On representable matroids having neither U2,5- nor U3,5-minors CHARLES SEMPLE AND GEOFF WHITTLE 377 Skeletal rigidity of p.1.-spheres TIONG-SENG TAY 387 The Coxeter matroids of Gelfand et al. NEILL. WHITE 401 Open problems 411

v Preface

With his 1935 paper On the abstract properties of linear dependence, founded the theory of matroids. The richness of Whitney's work can be attributed in part to the variety of fields from which he drew inspiration, including algebra, geometry, and . Since Whitney's paper, numerous authors have recognized the natural occurrence of matroids in a wide diversity of areas, and the interplay between matroid theory and other fields has flourished. This volume contains the proceedings of the 1995 AMS-IMS-SIAM Joint Sum- mer Research Conference Matroid Theory, held at the University of Washington, Seattle, from July 2 until July 6, 1995. The conference focused on four major areas within matroid theory and its fields of application: the critical problem; structure theory and connectivity; rigidity and discrete applied geometry; and oriented ma- troids and arrangements of hyperplanes. The conference was structured to achieve the dual goals of fostering greater interaction among researchers in different spe- cialties and presenting the latest research within each specialty. Each of the four chosen areas was surveyed by a leading researcher in the specialty. This volume begins with survey papers by Joseph Kung, James Oxley, and Walter Whiteley. These papers expand on the presentations made at the conference and blend the background material needed to make the papers accessible with numerous new results and research questions. Gunter Ziegler's survey of oriented matroids and arrangements of hyperplanes will be developed into a separate book. The survey lectures at the conference were complemented by more specialized talks and poster presentations. Papers resulting from these appear after the survey papers. The book concludes with a chapter of: open problems. We hope that the contents of this volume will be interesting to a wide audience, and will advance further developments in matroid theory and its applications. We thank the National Science Foundation for its financial support and the American Mathematical Society for its organizational support. Wayne Drady, of the AMS meetings and conferences department, did an excellent job with local arrangements, bookkeeping, and on-site assistance; Donna Harmon, of the AMS publications department, provided valuable guidance throughout the publication process. We are grateful to the referees. Finally, we thank those whose contributions are the core of this project: those who gave lectures and poster presentations at the conference, and those who contributed to this volume.

Joseph E. Bonin James G. Oxley Brigitte Servatius

vii Contemporary Mathematics Volume 197, 1996

Conference Program

Sunday, July 2 9:00-10:00 James Oxley, Structure theory and connectivity for matroids. Part I. 10:30-11:30 James Oxley, Structure theory and connectivity for matroids. Part II. 2:00-2:45 Geoff Whittle, Ternary matroids representable over other fields. 3:00-3:45 Dirk Vertigan, Matroid minors and structure. 4:00-4:45 Neil Robertson* and Jack Dharmatilake, Binary matroids of branch-width three. 7:00-8:30 Poster Session 1 o Gary Gordon* and Elizabeth McMahon, An algorithmic approach to activities for matroids, , and . o Arthur Hobbs, Uniformly dense matroids. o Sandra Kingan, A generalization of a graph result by D. W. Hall. o John Leo, On coefficients of the . o Allan Mills, Matroid reconstruction and relaxation.

Monday, July 3 9:00-10:00 Walter Whiteley, Matroids in discrete applied geometry. Part I: rigidity of frameworks. 10:30-11:30 Walter Whiteley. Matroids in discrete applied geometry. Part II: splines, pictures, and CAD. 2:00-2:45 Jack E. Graver, The matroid operator. 3:00-3:45 Robert Connelly, Is global rigidity a generic property'? 4:00-4:45 T. S. Tay, Rigidity and the number of faces of polytopes. 7:00-8:30 Poster Session 2 o Laura Anderson, Combinatorial differential manifolds. o Seth Chaiken, Oriented matroid pairs: theory and an electric application. o Camille C. Cooper, Bracing zonahedral frameworks. o Robert Jamison, Monotactic matroids. o Andras Recski, Some recent results on the combinatorial properties of square and cubic grids. o Paul Terwilliger, Quantum matroids.

@1996 American Mathematical Society ix X CONFERENCE PROGRAM

Tuesday, July 4 9:00-10:00 Henry Crapo, Tilings by related zonotopes, with a fresh look at oriented matroids. 10:30-11:30 Thomas Brylawski, A matroid theory generalization of the contravariant form in representation theory.

Wednesday, July 5 9:00-10:00 Joseph P. S. Kung, The critical problem: past and future. Part I: history and survey. 10:30-11:30 Joseph P. S. Kung, The critical problem: past and future. Part II: problems and methods. 2:00-2:45 Dominic Welsh, Randomised algorithms for computing numerical invariants. 3:00-3:45 Neil White, Matroids generalized to Coxeter groups - the (W, P)-matroids of Gelfand et al. 7:00-8:30 Poster Session 3 o Laura Anderson* and Rephael Wenger, Oriented matroids and hyperplane transversals. o Manoj K. Chari, On topology and enumeration in matroids. o Hong-Jian Lai* and Zhi-Hong Chen, Connectivity of matroids and bicircular matroids. o William Miller, Reconstructing matroids from Tutte poly- nomials. o Charles Semple, Matroids representable over subgroups of fields. o Haidong Wu, On contractible and vertically contractible ele- ments in 3-connected matroids and graphs. Thursday, July 6 9:00-10:00 Gi.inter M. Ziegler, Real arrangements, complex arrangements, and their {oriented) matroids. Part I. 10:30-11:30 Gi.inter M. Ziegler, Real arrangements, complex arrangements, and their (oriented) matroids. Part II. 2:00-2:45 Ji.irgen Richter-Gebert, Non realizability proofs for (oriented) matroids and automatic theorem proving. 3:00-3:45 Michael J. Falk, Applications of matroid theory to generalized hypergeometric functions. 4:00-4:45 Paul Edelman, Discriminantal arrangements, free arrange- ments, and enumeration. Contemporary Mathematics Volume 197, 1996

Participants List for Matroid Theory July 2-6, 1995, University of Washington, Seattle

Ian Affleck Camille Cooper Simon Fraser University York University Canada Canada

Laura Anderson Henry Crapo Indiana University at Bloomington INRIA USA France

Kenneth Bogart Thomas Dowling Dartmouth College Ohio State University USA USA

Joseph Bonin Paul Edelman The George Washington University University of Minnesota USA USA

Thomas Brylawski Michael Falk Univ. of North Carolina, Chapel Hill Northern Arizona University USA USA

Heidi Burgiel Luis Goddyn University of Washington Simon Fraser University USA Canada

Seth Chaiken Gary Gordon SUNY at Albany Lafayette College USA USA

Manoj Chari Jack Graver Louisiana State University Syracuse University USA USA

Robert Connelly Arthur Hobbs Cornell University Texas A & M University USA USA

@1996 American Mathematical Society xi xii PARTICIPANT LIST FOR MATROID THEORY

Robert Jamison Jiirgen Richter-Gebert Clemson University Technische Universitat Berlin USA Germany

Andre Kezdy Neil Robertson University of Louisville Ohio State University USA USA

Sandra Kingan Charles Semple Oakland University Victoria University of Wellington USA New Zealand

Joseph Kung Brigitte Servatius University of North Texas Worcester Polytechnic Institute USA USA

Hong-Jian Lai Tiong-Seng Tay West Virginia University National University of Singapore USA Singapore

John Leo Paul Terwilliger Louisiana State University University of Wisconsin USA USA

Jennifer McNulty Dirk Vertigan University of Montana Louisiana State University USA USA

William Miller Dominic Welsh The George Washington University Mathematical Institute, Oxford USA England

Allan Mills Neil White Louisiana State University University of Florida USA USA

James Oxley Walter Whiteley Louisiana State University York University USA Canada

Mark Purtill Geoffrey Whittle Texas A & M University-Kingsville Victoria University of Wellington USA New Zealand

Andras Recski Haidong Wu Technical University of Budapest Southern University Hungary USA

Talmage James Reid Gunter Ziegler University of Mississippi Technische Universitat Berlin USA Germany Selected Titles in This Series {Continued from the front of this publication)

169 William Abikoff, Joan S. Birman, and Kathryn Kuiken, Editors, The mathematical legacy of Wilhelm Magnus, 1994 168 Gary L. Mullen and Peter Jau-Shyong Shiue, Editors, Finite fields: Theory, applications, and algorithms, 1994 167 RobertS. Doran, Editor, C*-algebras: 1943-1993, 1994 166 George E. Andrews, David M. Bressoud, and L. Alayne Parson, Editors, The Rademacher legacy to mathematics, 1994 165 Barry Mazur and Glenn Stevens, Editors, p-adic monodromy and the Birch and Swinnerton-Dyer conjecture, 1994 164 Cameron Gordon, Yoav Moriah, and Bronislaw Wajnryb, Editors, Geometric topology, 1994 163 Zhong-Ci Shi and Chung-Chun Yang, Editors, Computational mathematics in China, 1994 162 Ciro Ciliberto, E. Laura Livorni, and Andrew J. Sommese, Editors, Classification of algebraic varieties, 1994 161 Paul A. Schweitzer, S. J., Steven Hurder, Nathan Moreira dos Santos, and Jose Luis Arraut, Editors, Differential topology, foliations, and group actions, 1994 160 Niky Kamran and Peter J. Olver, Editors, Lie algebras, cohomology, and new applications to quantum mechanics, 1994 159 William J. Heinzer, Craig L. Huneke, and Judith D. Sally, Editors, Commutative algebra: Syzygies, multiplicities, and birational algebra, 1994 158 Eric M. Friedlander and Mark E. Mahowald, Editors, Topology and representation theory, 1994 157 Alfl.o Quarteroni, Jacques Periaux, Yuri A. Kuznetsov, and Olof B. Widlund, Editors, Domain decomposition methods in science and engineering, 1994 156 Steven R. Givant, The structure of relation algebras generated by relativizations, 1994 155 William B. Jacob, Tsit-Yuen Lam, and Robert 0. Robson, Editors, Recent advances in real algebraic geometry and quadratic forms, 1994 154 Michael Eastwood, Joseph Wolf, and Roger Zierau, Editors, The Penrose transform and analytic cohomology in representation theory, 1993 153 RichardS. Elman, Murray M. Schacher, and V. S. Varadarajan, Editors, Linear algebraic groups and their representations, 1993 152 Christopher K. McCord, Editor, Nielsen theory and dynamical systems, 1993 151 Matatyahu Rubin, The reconstruction of trees from their automorphism groups, 1993 150 Carl-Friedrich Biidigheimer and Richard M. Hain, Editors, Mapping class groups and moduli spaces of Riemann surfaces, 1993 149 Harry Cohn, Editor, Doeblin and modern probability, 1993 148 Jeffrey Fox and Peter Haskell, Editors, Index theory and operator algebras, 1993 147 Neil Robertson and Paul Seymour, Editors, Graph structure theory, 1993 146 Martin C. Tangora, Editor, Algebraic topology, 1993 145 JeJrrey Adams, Rebecca Herb, Stephen Kudla, Jian-Shu Li, Ron Lipsman, and Jonathan Rosenberg, Editors, Representation theory of groups and algebras, 1993 144 Bor-Luh Lin and William B. Johnson, Editors, Banach spaces, 1993 143 Marvin Knopp and Mark Sheingorn, Editors, A tribute to Emil Grosswald: Number theory and related analysis, 1993 142 Chung-Chun Yang and Sheng Gong, Editors, Several complex variables in China, 1993 141 A. Y. Cheer and C. P. van Dam, Editors, Fluid dynamics in biology, 1993

(See the AMS catalog for earlier titles) Matroid Theory Joseph E. Bonin, James G. Oxley, and Brigitte Servatius, Editors This volume contains the proceedings of the 1995 AMS-IMS-SIAM Joint Sum- mer Research Conference on Matroid Theory held at the University of Washington, Seattle. The book features three comprehensive surveys that bring the reader to the forefront of research in matroid theory. Joseph Kung's encyclopedic treatment of the critical problem traces the de- velopment of this problem from its origins through its numerous links with other branches of mathematics to the current status of its many aspects. James Oxley's survey of the role of connectivity and structure theorems in matroid theory stresses the influence of the Wheels and Whirls Theorem of Thtte and the Splitter Theorem of Seymour. Walter Whiteley's article unifies applications of matroid theory to constrained geometrical systems, including the rigidity of bar-and-joint frameworks, parallel drawings, and splines. These widely accessible articles contain many new results and directions for further research and applications. The surveys are complemented by selected short research papers. The volume concludes with a chapter of open problems. Features: • Self-contained, accessible surveys of three active research areas in matroid theory. • Many new results. • Pointers to new research topics. • A chapter of open problems. • Mathematical applications. • Applications and connections to other disciplines, such as computer-aided design and electrical and structural engineering.

ISBN 0-8218-0508-8

9 780821 805084