Geometric Graphs and Arrangements: Some Chapters from Combinatorial Geometry

178 Pages·2004·9.78 MB·English

by  Prof. Dr. Stefan Felsner (auth.)

Stefan Felsner
Geometric Graphs
and Arrangements
Advanced Lectures
in Mathematics
Editorial Board
Prof. Dr. Martin Aigner, Freie Universitiit Berlin, Germany
Prof. Dr. Peter Gritzmann, Technische Universitat Mfinchen, Germany
Prof. Dr. Volker Mehrmann, Technische Universitat Berlin, Germany
Prof. Dr. Gisbert Wiistholz, ETH ZUrich, Switzerland
Introduction to Markov Chains
Ehrhard Behrends
~infilhrung In die Symplektische Geometrle
Rolf Berndt
Wavelets - Elne Einfilhrung
Christian Blatter
lattices and Codes
Wolfgang Ebeling
Local Analytic Geometry
Theo de Jong, Gerhard Pfister
Geometric Graphs and Arrangements
Stefan Felsner
Ruled Varieties
Gerd Fischer, Jens Piontkowski
Dlrac-Operatoren in der Rlemannschen Geometrie
Thomas Friedrich
Hypergeometric Summation
Wolfram Koepf
The Steiner Tree Problem
Hans Jiirgen Promel, Angelika Steger
The Basic Theory of Power Series
Jesus M. Ruiz
vieweg _________________
Stefan Felsner
Geometric Graphs
and Arrangements
Some Chapters from Combinatorial Geometry
aI
vleweg
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>.
Prof. Dr. Stefan Felsner
Technische Universitat Berlin
Institut fUr Mathematik (MA 6-1)
StraBe des 17. Juni 136
lO623 Berlin, Germany
[email protected]
Mathematics Subject Classification
MSC 2000: 05-01, 05ClO, 05C62, 52-01, 52ClO, 52C30, 52C42
First edition, February 2004
All rights reserved
© Friedr. Vieweg & Sohn Verlag/ GWV Fachverlage GmbH, Wiesbaden 2004
Vieweg is a company in the specialist publishing group Springer Science+Business Media.
www.vieweg.de
No part of this publication may be reproduced, stored in a retrieval system or
transmitted, mechanical, photocopying or otherwise without prior permission
of the copyright holder.
Cover design: Ulrike Weigel, www.CorporateDesignGroup.de
Printed on acid-free paper
ISBN -13:978-3-528-06972-8 e-ISBN-13:978-3-322-80303-0
DO I: 10.1007/978-3-322-80303-0
Preface
The body of this text is written. It remains to find some words to explain what to
expect in this book. A first attempt of characterizing the content could be:
MSC 2000: 05-01, 05(;10, 05C62, 52-01, 52C10, 52C30, 52C42.
In words: The questions posed and partly answered in this book are from the inter
section of graph theory and discrete geometry. The reader will meet some graph theory
with a geometric flavor and some combinatorial geometry of the plane. Though, the in
vestigations always start in the geometry of the plane it is sometimes appropriate to
pass on to higher dimensions to get a more global understanding of the structures under
investigation. This is the in Chapter 7, for example, when the study of triangulations of
a point configuration leads to the definition of secondary polytopes.
I like to think of the book as a collection which makes up a kind of bouquet. A bouquet
of problems, ideas and results, each of a special character and beauty, put together with
the intention that they supplement each other to form an interesting and appealing whole.
The main mathematical part of the text contains only few citations and references to
related material. These additional bits of information are provided in the last section of
each chapter, 'Notes and References'. On average the bibliography of a chapter contains
about thirty items. This is far from being a complete list of the relevant literature. The
intention is to just indicate the most valuable literature so that these sections can serve
as entry points for further studies. The text is supplemented by many figures to make the
material more attractive and help the reader get a sensual impression of the objects. In
some cases, I have confined the presentation to results which fall behind today's state of
the art. I wanted to emphasize the main ideas and stop before technical complexity starts
taking over. This strategy should make the mathematics accessibility to a relatively broad
audience including students of computer science, students of mathematics, instructors
and researchers.
The book can serve different purposes. It may be used as textbook for a course or as
a collection of material for a seminar. It should also be helpful to people who want to
learn something about specific themes. They may concentrate on single chapters because
all the chapters are self-contained and can be read as stand alone surveys.
Topics
Chapter 1. We introduce basic notion graph theory and explain what geometric and
topological graphs are. Planar graphs and some important theorems about them are
reviewed. The main results of this chapter are bounds for some extremal problems for
geometric graphs.
Chapter 2. We show that a 3-connected planar graph with f faces admits a convex drawing
on the (f - 1) x (f - 1) grid. The result is based on Schnyder woods, a special cover of
the edges of a 3-connected planar graph with three trees. Schnyder woods bring along
connections to geodesic embeddings of planar graphs and to the order dimension of planar
graphs and 3-polytopes.
VI Preface
Chapter 3. This is about non-planar graphs. How many crossing pairs of edges do we need
in any drawing of a given graph in the plane? The Crossing Lemma provides a bound
and has beautiful applications to deep extremal problems. We explain some of them.
Chapter 4. Let P be a configuration of n points in the plane. A k-set of P is a subset S
of k points of P which can be separated from the complement S = P \ S by a line. The
notorious k-set problem of discrete geometry asks for asymptotic bounds of this number
as a function of n. We present bounds, Welzl's generalization of the Lovasz Lemma
to higher dimensions and close with the surprisingly related problem of bounding the
rectilinear crossing number of complete graphs from below.
Chapter 5. This chapter contains selected results from the extremal theory for configura
tions of points and arrangements of lines. The main results are bounds for the number of
ordinary lines of a point configuration and for the number of triangles of an arrangement.
Chapter 6. Compared to arrangements of lines, arrangements of pseudolines have the
advantage that they can be nicely encoded by combinatorial data. We introduce several
combinatorial representations and prove relations between them. For each representation
we give an applications which makes use of specific properties. The encoding by triangle
orientations has a natural generalization which leads to higher Bruhat orders.
Chapter 7. In this chapter we study triangulations of a point configuration. The flip
operation allows to move between different triangulations. The Delaunay triangulation is
investigated as a special element in the graph of triangulations. This graph is shown to
be related to the skeleton graph of the secondary polytope. In the special case of a point
configuration in convex position they coincide. In this case, we make use of hyperbolic
geometry to get a lower bound for the diameter of the graph of triangulations.
Chapter 8. Rigidity allows a different view to geometric graphs. We introduce rigidity the
ory and prove three characterizations of minimal generically rigid graphs in the plane.
Pseudotriangulations are shown to be the planar minimal generically rigid graphs in the
plane. The set of pseudotriangulations with vertices embedded in a fixed point config
uration P has a nice structure. There is a notion of flip that allows to move between
different pseudotriangulations. The flip-graph is a connected graph and it turns out that
it is the skeleton graph of a polytope. The beautiful theory finds a surprising application
in the Carpenter's Rule Problem.
The selection of topics is clearly governed by my personal taste. This is a drawback,
because your taste is likely to differ from mine, at least in some details. The advantage
is that though there are several books on related topics each of these books is clearly
distinguishable by its style and content. In the spirit of 'Customers who bought this book
also bought' I recommend the following four books:
• J. MATOUSEK • G. M. ZIEGLER
Lectures on Discrete Geometry Lectures on Polytopes
Graduate Texts in Math. 212 Graduate Texts in Math. 152
Springer-Verlag, 2002. Springer-Verlag, 1994 .
• J. PACH AND P. K. AGARWAL • H. EDELSBRUNNER
Combinatorial Geometry Geometry and Topology for
John Wiley & Sons, 1995. Mesh Generation,
Cambridge University Press, 2001.
VII
Feedback
There will be something wrong. You may find errors of different nature. Inadvertently,
I may not have given proper credit for certain contribution. You may know of relevant
work that I have overlooked or you may have additional comments. In all these cases:
Please let me know .
•
felsner~ath.tu-berlin.de
You can find a list of errata and a collection of comments and pointers related to the
book at the following web-location:
• http://www.math.tu-berlin.de/-felsner/gga-book.html
Acknowledgments
There are sine qua non conditions for the existence of this book: The work of a vast
number of mathematicians who laid out the ground and produced what I am retelling.
Friends and family of mine, in particular Diana and my parents. without their persistent
trust -blind or not- I would not have been able to finish this task. I am truly grateful to
all of them.
In Berlin we have a wonderful environment for discrete mathematics. While working
on this book, I had the privilege of working in teams of discrete mathematicians at the
Freie Universitat and at the Technische Unversitat. I want to thank the nice people in
these groups for providing me with such a friendly and supportive 'ambiente'.
I have been involved in the European graduate college 'Combinatorics, Geometry and
Computing' and in the European network COMBSTRU and I am glad to acknowledge
support from their side.
I'm grateful to Martin Aigner, he furthered this project with advice and the periodically
renewed question: "What about your book Stefan?" . For valuable discussions and for help
by way of comments and corrections I want to thank Patrick Baier, Peter Brafi, Frank
Hoffmann, Klaus Kriegel, Ezra Miller, Walter Morris, Janos Pach, Gunter Rote, Sarah
Renkl, Volker Schulz, Ileana Streinu, Tom Trotter, Pavel Valtr, Uli Wagner, Helmut Weil,
Emo Welzl and Gunter Ziegler.
Berlin, December 2003
Stefan Felsner
Contents
Preface v
Contents IX
1 Geometric Graphs: Turan Problems 1
1.1 What is a Geometric Graph? ...... . 1
1.2 Fundamental Concepts in Graph Theory. 2
1.3 Planar Graphs . . . . . . . . . . . . . . . 3
1.4 Outerplanar Graphs and Convex Geometric Graphs 6
1.5 Geometric Graphs without (k + I)-Pairwise Disjoint Edges 7
1.6 Geometric Graphs without Parallel Edges 10
1. 7 Notes and References ............... . 13
2 Schnyder Woods or How to Draw a Planar Graph? 17
2.1 Schnyder Labelings and Woods .... . 17
2.2 Regions and Coordinates ....... . 23
2.3 Geodesic Embeddings of Planar Graphs 24
2.4 Dual Schnyder Woods . . . . . . 28
2.5 Order Dimension of 3-Polytopes 30
2.6 Existence of Schnyder Labelings 35
2.7 Notes and References ..... . 37
3 Topological Graphs: Crossing Lemma and Applications 43
3.1 Crossing Numbers ........ . 43
3.2 Bounds for the Crossing Number ..... . 44
3.3 Improving the Crossing Constant . . . . . . 46
3.4 Crossing Numbers and Incidence Problems 48
3.5 Notes and References ........... . 50
4 k-Sets and k-Facets 53
4.1 k-Sets in the Plane . 53
4.2 Beyond the Plane ........... . 58
4.3 The Rectilinear Crossing Number of K n 61
4.4 Notes and References ......... . 66
5 Combinatorial Problems for Sets of Points and Lines 69
5.1 Arrangements, Planes, Duality ...... . 69
5.2 Sylvester's Problem ............ . 72
5.3 How many Lines are Spanned by n Points? 76
5.4 Triangles in Arrangements. 78
5.5 Notes and References ........... . 84
X Contents
6 Combinatorial Representations of Arrangements of Pseudolines 87
6.1 Marked Arrangements and Sweeps. . . . . 87
6.2 Allowable Sequences and Wiring Diagrams 91
6.3 Local Sequences . 95
6.4 Zonotopal THings ...... 100
6.5 Triangle Signs ........ 105
6.6 Signotopes and their Orders . 108
6.7 Notes and References 111
7 Triangulations and Flips 114
7.1 Degrees in the Flip-Graph. . . . . . . . . . . . . 114
7.2 Delaunay Triangulations. . . . . . . . . . . . . . 117
7.3 Regular Triangulations and Secondary Polytopes 120
7.4 The Associahedron and Catalan families. . . . 123
7.5 The Diameter of fin and Hyperbolic Geometry 125
7.6 Notes and References .............. 129
8 Rigidity and Pseudotriangulations 131
8.1 Rigidity, Motion and Stress . . . . . . . . . 131
8.2 Pseudotriangles and Pseudotriangulations . 139
8.3 Expansive Motions . . . . . . . . . . . . . . 143
8.4 The Polyhedron of of Pointed Pseudotriangulations 144
8.5 Expansive Motions and Straightening Linkages 148
8.6 Notes and References ................. 149
Bibliography 151
Index 165
1 Geometric Graphs: Turan Problems
The first part of this chapter collects rather elementary and well known material: We
start with definitions of geometric and topological graphs, rush through basic notions
from graph theory and report on facts about planar graphs. Beginning with Section 1.4
we discuss problems from the extremal theory for geometric graphs. That is, we deal with
questions of Thran type: How many edges can a geometric graph avoiding a specified
configuration of edges have?
1.1 What is a Geometric Graph?
As usual in texts dealing with graph theory, we shall begin by defining a graph G to be
a pair (V, E) where V is the set of vertices, E is the set of edges {v, w} each joining two
vertices v, w E V. To illustrate the definition, let us look at some pictures.
Petersen
graph
Figure 1.1 A little gallery of graphs, the cube graph Q3 is shown with two drawings.
A geometric graph is a graph G = (V, E) drawn in the plane with straight edges. That
is, V is a set of points in the plane and E is a set of line segments with endpoints in
V. For convenience, we often assume that the points of V are in general position, i.e.,
no three points of V are collinear. Pictures of graphs, as Figure 1.1, display geometric
graphs. Hence, the study of geometric graphs can be viewed as the study of straight line
drawings of a graph.
A topological graph is a graph drawn in the plane such that edges are represented by
Jordan curves with the property that any two of these curves share at most one point.
Obviously, geometric graphs are a subclass of topological graphs
In general, a drawing of a graph G will display crossings of edges that are disjoint
as edges of G. Many interesting questions about geometric and topological graphs are
S. Felsner, Geometric Graphs and Arrangements
© Friedr. Vieweg & Sohn Verlag/GWV Fachverlage GmbH, Wiesbaden 2004