Graphs and Order: The Role of Graphs in the Theory of Ordered Sets and Its Applications

797 Pages·1985·34.142 MB·English

by  David Kelly (auth.), Ivan Rival (eds.)

Graphs and Order
The Role of Graphs in the Theory of Ordered Sets and Its Applications
NATO ASI Series
Advanced Science Institutes Series
A series presenting the results of activities sponsored by the NATO Science Committee,
which aims at the dissemination of advanced scientific and technological knowledge,
with a view to strengthening links between scientific communities.
The series is published by an international board of publishers in conjunction with the
NATO Scientific Affairs Division
A Life Sciences Plenum Publishing Corporation
B Physics London and New York
C Mathematical D. Reidel Publishing Company
and Physical Sciences Dordrecht, Boston and Lancaster
D Behavioural and Social Sciences Martinus Nijhoff Publishers
E Engineering and The Hague, Boston and Lancaster
Materials Sciences
F Computer and Systems Sciences Springer-Verlag
G Ecological Sciences Berlin, Heidelberg, New York and Tokyo
Series C: Mathematical and Physical Sciences Vol. 147
Graphs and Order
The Role of Graphs in the Theory of Ordered Sets
and Its Applications
edited by
Ivan Rival
Department of Mathematics and Statistics,
The University of Calgary,
Calgary, Alberta, Canada
D. Reidel Publishing Company
Dordrecht / Boston I Lancaster
Published in cooperation with NATO Scientific Affairs Division
Proceedings of the NATO Advanced Study Institute on
Graphs and Order
Banff,Canada
May 18-31,1984
Library of Congress Cataloging in Publication Data
NATO Advanced Study Institute on Graphs and Order
(1984 : Banff, Alta.)
Graps and order.
(NATO ASI series. Series C, Mathematical and physical sciences; v. 147)
"Published in cooperation with NATO Scientific Affairs Division."
"Proceedings of the NATO Advanced Study Institute on Graphs and Order,
Banff, Canada, May 18-31, 1984"-T.p. verso.
Bibliography: p.
Includes index.
1. Graph theory-Congresses. 2. Ordered sets-Congresses. I. Rival, Ivan, 1947-
II. North Atlantic Treaty Organization. Scientific Affairs Division. III. Title.
IV. Series: NATO ASI series. Series C, Mathematical and physical sciences; no. 147.
QA 166.N38 1984 511'.5 84-27696
ISBN-13: 978-94-010-8848-0 e-ISBN-13: 978-94-009-5315-4
001: 10.1007/978-94-009-5315-4
Published by D. Reidel Publishing Company
p. O. Box 17, 3300 AA Dordrecht, Holland
Sold and distributed in the U.sA and Canada
by Kluwer Academic Publishers,
190 Old Derby Street, Hingham, MA 02043, U.SA
In all other countries, sold and distributed
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D. Reidel Publishing Company is a member of the Kluwer Academic Publishers Group
All Rights Reserved
©1985 by D. Reidel Publishing Company, Dordrecht, Holland.
Softcover reprint of the hardcover 15t edition 1985
No part of the material protected by this copyright notice may be reproduced or utilized
in any form or by any means, electronic or mechanical, including photocopying, recording
or by any information storage and retrieval system, without written permission from the
copyright owner.
CONTENTS
PREFACE
x~
PARTICIPANTS xiii
SCIENTIFIC PROGRAMME xv
PART 1. GRAPH REPRESENTATIONS
Comparability graphs
D. Kelly 3
Algorithmic aspects of comparability graphs and
interval graphs
R.H. Mohring 41
The diagram
I. Rival 103
PART II. SCHEDULING, SEARCHING AND SORTING
The information theoretic bound for problems on
ordered sets and graphs
M.E. Saks 137
Sorting and graphs
B. Bollobas and P. Hell 169
A graph-theoretic approach to the jump-number problem
M.M. Syslo 185
Acyclic subdigraphs and linear orderings: polytopes,
facets, and a cutting plane algorithm
M. Grotschel, M. Junger and G. Reinelt 217
PART III. EXTREMAL ORDERS
Parameters of partial orders and graphs: packing,
covering and representation
D.B. West 267
vi CONTENTS
Graphs and orders in Ramsey theory and
in dimension theory
M. Paoli, W.T. Trotter, Jr. and J.W. Walker 351
Ordered ranked posets, representations of
integers and inequalities from extremal
poset problems
D.E. Daykin 395
PART IV. SOCIAL SCIENCES
Issues in the theory of uniqueness in
measurement
F.S. Roberts 415
PART V. DECOMPOSITION
Path-partitions in directed graphs and posets
C. Berge 447
PARTVr. RECURSION
Recursion theoretic aspects of graphs and
orders
J.H. Schmerl 467
PART VII. WELL-QUASI-ORDERING
Basic wqo- and bqo-theory
E.C. Milner 487
Applications of well quasi-ordering and
better quasi-ordering
M. Pouzet 503
PART VIII. PROBLEM SESSIONS
Introduction 522
Enumeration 523
The diagram 529
Ramsey theory 535
CONTENTS vii
Sorting, scheduling, computing, operations
research, and social science 543
Structure theories 549
Infinite sets and infinite graphs 559
Souvenir session 567
Order-preserving and edge-preserving maps 571
Miscellaneous 581
PART IX. A BIBLIOGRAPHY
Introduction 592
PART X. INDEX 779
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(LEFT TO RIGHT) O. Pretzel, F. Farmer, I'LT. Trotter, M. Fournier, R. Hoodrow, P. HinkleJ. H. Kierstead, P. Baayen, S. Grantham, Stahl, D. Duffus, P. Hell, J. Brown. J. ROH (LEFT TO RIGHT) D. Skilton, Almeida, S. Thomeier, P. Vestergaard, J. Longyear, C. B. Sands, H. Li, E. Graczynska-Wronska, D. Kelly, Mrs. G. Daykin, D. Daykin, M.H. EI-Zah OH (LEFT TO RIGHT) S. Todorcevic, R. Nowakowski, J.-M. PIa, M. Habib, N. Zaguia, P. FishM. Golombic, R. Mohring, M. Syslo, H. Pouzet, J. Constantin, H. JUnger, E. Milner, }1aste ROH (LEFT TO RIGHT) I. Broere, Albertson, A. Pitkin, M. Saks, D. West, A. Kotzi?, M. }!. A. Vince, B. Hunson, R. Simion, J. Pfaltz, K.-M. Koh, E. Harzheim. OH (LEFT TO RIGHT) Mrs. G. Broere, Master D. Broere, Haster A. Steiner, Hrs. J. Steiner, G. Steiner, C. Colbourn, J. Larson, Rival, Master D. Rival, Mrs. H. Rival, Master R. 1. D. Klarner, E. Corominas, K. Prikry, Mrs. S. EI-Zahar, Miss H. EI-Zahar. apticipants not in pictupe: C. Berge, A. Bialostocki, S. Fisk, Z. Furedi, A. Hajnal, D. D. Higgs, H. Klamkin, B. Korte, E. Lawler, D. Miller, J. Pach, P. Ribenboim, F. Roberts, N. Sauer, J. Schmerl, H. Weiss
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TOP FOUR THIR SECO FRON Othe
PREFACE
This volume contains the accounts of the principal survey papers
presented at GRAPHS and ORDER, held at Banff, Canada from May 18 to
May 31, 1984. This conference was supported by grants from the N.A.T.O.
Advanced Study Institute programme, the Natural Sciences and Engineering
Research Council of Canada and the University of Calgary. We are
grateful for all of this considerable support.
Almost fifty years ago the first Symposium on Lattice Theory was
held in Charlottesville, U.S.A. On that occasion the principal lectures
were delivered by G. Birkhoff, O. Ore and M.H. Stone. In those days
the theory of ordered sets was thought to be a vigorous relative of
group theory.
Some twenty-five years ago the Symposium on Partially Ordered Sets
and Lattice Theory was held in Monterey, U.S.A. Among the principal
speakers at that meeting were R.P. Dilworth, B. Jonsson, A. Tarski and
G. Birkhoff. Lattice theory had turned inward: it was concerned
primarily with problems about lattices themselves. As a matter of fact
the problems that were then posed have, by now, in many instances, been
completely solved.
Four years ago the Symposium on Ordered Sets was held at Banff,
Canada. Our charge was to document the vitality of ordered sets. The
problems posed then were more purely order-theoretic than at any of
the earlier meetings. The subject had developed a wider scope due to
important links established with modern combinatorial theory. That
meeting included lectures by several of the earliest pioneers, including
B. Jonsson, R.P. Dilworth and G. Birkhoff.
As a matter of fact the present conference was inspired by what
seemed to me an important omission in the programme of the Symposium
on Ordered Sets (1981). The role of graph theory within the theory of
ordered sets had not then been properly delineated. GRAPHS and ORDER
(1984) was to document the role of graphs in the theory of ordered sets
and its applications.
xi