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Table of contents :
Title Page
Preface
Organization
Program Committee
Organizing Committee
Table of Contents
The Anatomy of a Geometric Algorithm?
Turn-Regularity and Planar Orthogonal
Drawings*
Introduction
Preliminaries
Basic Definitions
Switch-Regularity
Turn-Regularity and Switch-Regularity
Orthogonal Relations
Turn-Regularity
Turn-Regularity and Switch-Regularity
Orientations and Paths
Turn-Regularity and Orthogonal Relations
Turn-Regularity and Drawing Algorithms
Experiments
Compaction Heuristics
Test Suite and Experimental Results
Conclusions and Future Work
Combining Graph Labeling and Compaction*
Introduction
Constraint Graphs
Modeling the Labels
Optimal Graph Labeling
Conclusions and Future Plans
Almost Bend-Optimal Planar Orthogonal
Drawings of Biconnected Degree-3 Planar
Graphs in Quadratic Time*
Introduction
Spirality and Polar Drawings
Drawing Triconnected Cubic Plane Graphs with Minimum Bends
SPQR Tree
Main Theorem
Fully Dynamic 3-Dimensional Orthogonal Graph
Drawing*
Introduction and Previous Work
Drawing Graphs in $O(n^2)$ Volume with 6 Bends
Introduction and Definitions
Proof of Correctness
Dynamic Case
Variants
Achieving 5 Bends per Edge
Spiral Layout
Self-Loops
Implementation and Experimental Results
Conclusion and Open Problems
Appendix: Routings
An E log E Line Crossing Algorithm for Levelled
Graphs
Introduction
Previous Work
Leveled Graphs and Layout
Leveled Graphs and Intersections
The LevelCross Algorithm
Implementation Results
Conclusions
Level Planar Embedding in Linear Time*
Introduction
Preliminaries
Concept of the Algorithm
Augmentation
Remarks
Higres - Visualization System for Clustered
Graphs and Graph Algorithms*
Introduction
Clustered Graphs in Higres
Visualization
User Interface
Algorithms Animation
Conclusion
Partitioning Approach to Visualization of Large
Graphs
Approaches to Clustering in Graphs
Cores
Core Decomposition
Clusterings in Graphs
Example -- World Trade Graph
Conclusion
Graph Clustering Using Distance-k Cliques
Software Demonstration
Introduction
Clustering Using Distance-k Cliques
Description of the Algorithms
Analysis of Algorithms
Experimental Results and Discussions
A Framework for Circular Drawings
of Networks*
Introduction
Circular Drawings of Biconnected Graphs
Circular Drawings of Nonbiconnected Graphs
Drawings of Nonbiconnected Graphs
Algorithm CIRCULAR - with Radial
Implementation and Experiments
Drawing Planar Graphs with Circular Arcs
Introduction
Prior Related Work
Our Results
Algorithm
Vertex Joint Box
The Invariants
The Shifting Set
The Construction
Drawing with Circular Arcs
Drawing Graphs in the Hyperbolic Plane
Introduction
The Poincar{{accent 19 e}} Half-Plane
The Unit Disk Model of the Hyperbolic Plane
Hyperbolic Primal Dual Circle Packings
Drawing Primitives
Circles
Hyperbolic Line Segments
Drawing in the Hyperbolic Plane
Examples
Graph Planarity and Related Topics
Basics
Testing Planarity in Linear Time
Schnyder's Theorem and Drawing on a Grid
Colin de Verdiere's Parameter
Separators
The Two Paths Problem
Pfaffian Orientations
Linkless Embeddings
A Digression
The Four-Color Theorem
Outline of a Proof of the 4CT
Grid Drawings of Four-Connected Plane Graphs
Introduction
Main Theorem
Drawing G'
Graph Embedding with Topological
Cycle-Constraints*
Introduction
Preliminaries
Cycle-Constraints
Restricted Cycle-Constraints
Decomposition Trees
Embedding Algorithm
Embedding Vertices at Points:
Few Bends Suffice for Planar Graphs
Introduction
A Basic Technique
The General Case
The Lower Bound
The NP-Completeness Result
Discussion
The Constrained Crossing Minimization
Problem
Introduction
Constrained Crossing Minimization and Shortest Crossing Walks
An ILP for Trails
An ILP for the Shortest Crossing Trails Problem
The Shortest Crossing Walks Problem
Computational Experiments and Results
Planarity-Preserving Clustering and Embedding
for Large Planar Graphs*
Introduction
Prior Related Work on Clustered Graph Drawing
Our Results
Hierarchical Embedding of Planar Graphs
Edge Contraction and Separating Triangles
Simple Matching in Maximally Planar Graphs
Algorithm and Analysis
Constructing the Embedding
An Algorithm for Drawing Compound Graphs
Introduction
Definitions
Algorithm for Drawing Compound Graphs
Description of the NUAGE Algorithm
Step 1: Construction of a Nested Graph Associated with a Compound Graph
Step 2: Drawing of the Nested Graph
Refinement Technique
The Vertex-Exchange Graph: A New Concept
for Multi-level Crossing Minimisation
Introduction
The Vertex-Exchange Graph
Definition and Properties
Odd-Labelled Cycles
Level Planarity Testing by the Vertex-Exchange Graph
Layout Calculation of a Level Planar Graph
Crossing Minimisation
Crossing Number Bounds
Conclusions
Using Sifting for k-Layer Straightline Crossing
Minimization
Introduction
Algorithms Known from the Literature
Sifting for One Sided Crossing Minimization
Extending Sifting to k-Layered Directed Graphs
Conclusions
On 3-Layer Crossings and Pseudo Arrangements
Introduction
Basic Concepts and Notations
Arrangements and 3-Layer Drawings
Visualizing Algorithms for the Design and Analysis of
Survivable Networks
1 Introduction
2 Inside Drawing
3 Outside Drawing
4 Mixed Drawing
4.1 Basic idea
4.2 Algorithm
5 Conclusion and Open Problems
6 Acknowledgement
References
LayoutShow: A Signed Applet/Application for
Graph Drawing and Experimentation
Introduction
Features
Outline
The LayoutShow Applet
The Overall Design of the System
Bugs and Limitations
Centrality in Policy Network Drawings
Introduction
Policy Networks and Centrality
Centrality Drawings
Analyzing Local Drug Policies
Straight-Line Drawings of Protein Interactions
Introduction
Biological Context
Large Straight-Line Drawings
Evaluation
Further Work
Conclusions
Acknowledgements
Art of Drawing
Introduction
Drawing and Sketching
The Basic Scheme
Defining Sketches
Nature of Sketches
Eternal Style and Quality
Rarity and Inaccessibility
Summary (Of the First Part of the Lecture)
Appendix
An Heuristic for Graph Symmetry Detection
Introduction
Automorphisms and Symmetries
The Embedding Model
Euclidean Semi-Distances
Defining a Semi-Distance in a Graph
Embedding and Projecting the Graph on Planes
The Heuristic by Examples
All the Eigenvalues are Distinct
The First Eigenvalue has Multiplicity Two
General Case
More Pictures
Open Questions
Concluding Remarks
Isomorphic Subgraphs
Introduction
Basic Notions
NP-Completeness of IES and INS
Outerplanar Graphs
Planar Graphs
Isomorphic Subtrees
Open Questions and Outlook
Orthogonal and Quasi-upward Drawings
with Vertices of Prescribed Size*
Introduction
Preliminaries and Drawing Conventions
Computing Podavsnef Drawings
Experiments with GDToolkit
Computing Pqudavs Drawings
Multi-dimensional Orthogonal Graph Drawing
with Small Boxes
Introduction
The General Position Model
Determining Vertex Size
Port Assignment
Balanced Graph Layout
Layout-Based Algorithms
Routing-Based Algorithms
Geometric Realization of Simplicial Complexes
Introduction
The Complex of a $d$-Representation
Geometric Realizations of Simplicial Complexes
Geometric Realization of a $d$-Representation
Triangulation
Shellability of Standard Complexes
Visibility Representations of Complete Graphs
Introduction
General properties
Triangles
Open problems
Triangle-Free Planar Graphs as Segments
Intersection Graphs
Introduction
Convex Faces with Three Directions
Triangle-Free Graphs
Concluding Remarks
A Force-Directed Algorithm that Preserves Edge
Crossing Properties
Introduction
PrEd algorithm
Computation of forces
Computation of the amplitude of moves
Correctness
Further work
Conclusion
Rectangle of Influence Drawings of Graphs
without Filled 3-Cycles*
Introduction
Definitions
Rectangle of Influence Drawings for NF3-Graphs
Biconnected, Chordless, Internally Triangulated {{NF3-graphs} }
Extension to all NF3-Graphs
Conclusion
Voronoi Drawings of Trees?
Introduction
Preliminaries
Euclidean Voronoi Drawings of Trees
Point Sets in the Manhattan Metric
Manhattan Voronoi Drawings of Trees
Infinite Trees and the Future*
Introduction
The Framework and the Drawing Strategies
Experimental Results
Analysis
Analysis of Leftmost-Embedding Algorithm
Analysis of Central-Embedding Algorithm
Latour - A Tree Visualisation System
Introduction
Graph/Tree Layout
Hierarchical View
Radial View
Balloon View
Interaction and Navigation
Zoom, Pan, Fish--Eye
Complexity Visual Cues
Animation
Beyond Trees
Packed Forests
Dag's
Conclusions
Graph-Drawing Contest Report
Introduction
Winning Submissions
Category A
Category B
Category C
Category D
Observations and Conclusions
Hunting Down Graph B*
Orthogonal and Straight-Line Drawings of
Graphs with Succinct Representations*
Electronic Biochemical Pathways*
Author Index
Recommend Papers

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Lecture Notes in Computer Science Edited by G. Goos, J. Hartmanis and J. van Leeuwen

1731

3

Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo

Jan Kratochv´ıl (Ed.)

Graph Drawing 7th International Symposium, GD’99 ˇ r´ın Castle, Czech Republic Stiˇ September 15-19, 1999 Proceedings

13

Series Editors Gerhard Goos, Karlsruhe University, Germany Juris Hartmanis, Cornell University, NY, USA Jan van Leeuwen, Utrecht University, The Netherlands Volume Editor Jan Kratochv´ıl (Ed.) Charles University, Faculty of Mathematics and Physics School of Computer Science, Department of Applied Mathematics 118 00 Prague 1, Czech Republic E-mail: [email protected]

Cataloging-in-Publication data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Graph drawing : 7th international symposium ; proceedings / GD ’99, St´ır´ın Castle, Czech Republic, September 15 - 19, 1999 / Jan Kratochv´ıl (ed.). - Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Singapore ; Tokyo : Springer, 1999 (Lecture notes in computer science ; Vol. 1731) ISBN 3-540-66904-3

CR Subject Classification (1998): G.2, I.3, F.2 ISSN 0302-9743 ISBN 3-540-66904-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1999 Printed in Germany Typesetting: Camera-ready by author SPIN: 10704020 06/3142 – 5 4 3 2 1 0

Printed on acid-free paper

Preface

The range of issues considered in graph drawing includes algorithms, graph theory, geometry, topology, order theory, graphic languages, perception, applications, and practical systems. Much research is motivated by applications to systems for viewing and interacting with graphs. The interaction between theoretical advances and implemented solutions is an important part of the graph drawing field. The annually organized graph drawing symposium is a forum for researchers, practitioners, developers, and users working on all aspects of graph visualization and representations. The preceding symposia were held in Montreal (GD’98), Rome (GD’97), Berkeley (GD’96), Passau (GD’95), Princeton (GD’94), and Paris (GD’93). The Seventh International Symposium on Graph Drawing GD’99 was orgaˇ r´ın Castle, in the vicinity of Prague, Czech Republic. This baroque nized at Stiˇ castle recently restored as a hotel and conference center provided a secluded place for the participants, who made good use of the working atmosphere of the conference. In total the symposium had 83 registered participants from 16 countries. The program committee had a hard time choosing the accepted 38 contributions from the 59 submitted ones. Every paper was read by at least 4 program committee members, carefully evaluated, and the final decision was made after a strenuous e-mail discussion. The accepted papers show the field of graph drawing in its broadness and versatility, with contributions ranging from theoretical results on graph representations and theory of graph drawing algorithms to system demonstrations of graph drawing software and practical applications. The poster gallery included only two presentations this year, both presented in the proceedings. All three invited speakers provide strong links to the Prague school of Discrete Mathematics. Jiˇrı Matouˇsek (Charles University, Prague) delivered a talk that showed a sample of techniques and tools used in designing geometric algorithms. Robin Thomas (Georgia Institute of Technology, Atlanta) presented a survey about planar graphs and related graph classes, including an overview of their recent computer aided re-proof of the Four Color Theorem. Both of these talks clearly showed the multiple connections of mathematics and practical computer science. Finally Jaroslav Neˇsetˇril (Charles University, Prague), former advisor of both preceding speakers, presented a somewhat atypical talk. His multimedia show gave another dimension to the notion of ‘drawing’ and reflected the recent trends in the graph drawing area, where aestethic criteria continue to play an increasingly important role. The proceedings are structured to reflect the program of the conference. In particular, all three invited talks are illustrated in the proceedings either by reprints of the transparencies or by an extended abstract.

VI

Preface

The graph drawing contest was again the center of interest for the participants. This contest serves to monitor and to challenge the state of the art in the area of graph drawing. A contest report is included in the proceedings. Thanks to Joe Marks for organizing the contest and taking the time to put together the comprehensive report. As a novelty a special prize was awarded to the best student contribution. Six of the accepted submissions were co-authored solely by students, and the program committee voted the best among these. The winner was David Wood for his paper ‘Multi-Dimensional Orthogonal Graph Drawing with Small Boxes’. Apart from the professional program, the participants warmly accepted a conference concert stivin@stirin given by one of the best known contemporary Czech jazz musicians Jiˇr´ı Stiv´ın and rewarded his one hour performance with a long lasting applause. GD’99 was organized and sponsored by DIMATIA Charles University. This recently established Center for Discrete Mathematics, Theoretical Computer Science and Applications fosters and coordinates research in Discrete Mathematics in the Czech Republic, and also has strong international links. The center organizes series of workshops and hosts both short term visitors and long term postdoc programs. For more information see its Web page http://www.ms.mff.cuni.cz/acad/kam/dimatia/ . Many people have contributed to the success of GD’99. First of all the authors, who submitted and presented high quality papers and system demonstrations. Program committee members invested a lot of their time into reading, evaluating, and selecting the accepted papers. Thanks are due to all members of the organizing committee and to the volunteers (mostly students of Charles University). Special thanks to Jan Vondr´ak for organizing the poster gallery and to V´ıt Nov´ak for making the computers and equipment ready when they were needed. But most special thanks are due to Jiˇr´ı Fiala for managing the electronic submissions, collecting and proof-reading the accepted papers, and technically editing these proceedings, and to Anna Kotˇeˇsovcov´a from Conforg for excellent local organization. Without the help of these two special people the conference would have been just impossible. Finally I want to thank the industrial sponsors for their financial or other help: Velk´e Popovice Brewery, Bohemia Glass, Sahm, Mitsubishi Electric Research Laboratory, AT&T Labs, and Tom Sawyer Software. The financial support ˇ from DIMATIA was partially made available by Czech Research grants GACR 201/1999/0242 and Kontakt 1999/0338. The coming Graph Drawing ’00 will be organized by Joe Marks and Kathy Ryall in Williamsburg, Virginia, in September 2000. September 1999

Jan Kratochv´ıl

Preface

Organization

GD’99 was organized by the

Program Committee Giuseppe Di Battista (Rome) Franz Brandenburg (Passau) Hubert de Fraysseix (Paris) Emden Gansner (AT&T) Jan Kratochv´ıl (Chair, Prague) Bill Lenhart (Williamstown) Kim Marriott (Melbourne) Joe Marks (MERL)

Bojan Mohar (Ljubljana) Petra Mutzel (Saarbr¨ ucken) Takao Nishizeki (Tohoku) Kathy Ryall (Charlottesville) Ioannis Tollis (Dallas) Pavel Valtr (Prague) Sue Whitesides (Montreal)

Steering Committee Giuseppe Di Battista (Rome) Franz Brandenburg (Passau) Hubert de Fraysseix (Paris) Peter Eades (Newcastle) Jan Kratochv´ıl (Prague)

Joe Marks (MERL) Takao Nishizeki (Tohoku) Roberto Tamassia (Providence) Ioannis Tollis (Dallas) Sue Whitesides (Montreal)

Organizing Committee Jiˇr´ı Fiala V´ıt Janota Anna Kotˇeˇsovcov´a Jan Kratochv´ıl

Jaroslav Neˇsetˇril V´ıt Nov´ak Pavel Valtr

Volunteers ˇ asensk´a Hana C´ ˇ Frantiˇsek Cernohorsk´ y Alena Fialov´ a Tom´ aˇs Chudlarsk´ y Romana Jezdinsk´ a

Petra Kadlecov´a Daniel Kr´ al Jana Maxov´ a ˇ Jakub Simek Jan Vondr´ ak

VII

VIII

Preface

External Referees E. Dahlhaus W. Didimo G. Farr W. Feng C. Gutwenger M. Himsolt M.-L. Huang M. Kaufmann

G. Klau X. Lin G. Liotta K. Miura S. Nakano J. Neˇsetˇril P. Ossona de Mendez M. Patrignani

M. Pizzonia S. Rahman F. Schreiber R. Webber R. Weiskircher D. Wood T. Ziegler

Table of Contents

Invited Talk The Anatomy of a Geometric Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jiˇr´ı Matouˇsek (Charles University)

1

Orthogonality I Turn-Regularity and Planar Orthogonal Drawings . . . . . . . . . . . . . . . . . . . . . Stina S. Bridgeman, Roberto Tamassia, Luca Vismara (Brown University), Giuseppe Di Battista, Walter Didimo (Universit` a di Roma Tre) and Giuseppe Liotta (Universit` a di Perugia)

8

Combining Graph Labeling and Compaction . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Gunnar W. Klau, Petra Mutzel (Max–Planck–Institut f¨ ur Informatik) Almost Bend-Optimal Planar Orthogonal Drawings of Biconnected Degree-3 Planar Graphs in Quadratic Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Ashim Garg (State University of New Yorka at Buffalo), Giuseppe Liotta (Universita’ Di Perugia) Fully Dynamic 3-Dimensional Orthogonal Graph Drawing . . . . . . . . . . . . . . . 49 M. Closson, S. Gartshore, J. Johansen and S. K. Wismath (University of Lethbridge)

Levels I An E log E Line Crossing Algorithm for Levelled Graphs . . . . . . . . . . . . . . . . 59 Vance Waddle, Ashok Malhotra (IBM Thomas J. Watson Research Center) Level Planar Embedding in Linear Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Michael J¨ unger, Sebastian Leipert (Universit¨ at zu K¨ oln) Higres – Visualization System for Clustered Graphs and Graph Algorithms 82 Ivan A. Lisitsyn, Victor N. Kasyanov (A. P. Ershov’s Institute of Informatics Systems)

Clusters I Partitioning Approach to Visualization of Large Graphs . . . . . . . . . . . . . . . . 90 Vladimir Batagelj, Andrej Mrvar and Matjaˇz Zaverˇsnik (University of Ljubljana)

X

Table of Contents

Graph Clustering Using Distance-k Cliques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Jubin Edachery, Arunabha Sen (Arizona State University) and Franz J. Brandenburg (Universit¨ at Passau)

Drawing I A Framework for Circular Drawings of Networks . . . . . . . . . . . . . . . . . . . . . . . 107 Janet M. Six, Ioannis G. Tollis (The University of Texas) Drawing Planar Graphs with Circular Arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 C. C. Cheng, C. A. Duncan, M. T. Goodrich and S. G. Kobourov (The John Hopkins University) Drawing Graphs in the Hyperbolic Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Bojan Mohar (University of Ljubljana)

Invited Talk Graph Planarity and Related Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Robin Thomas (Georgia Institute of Technology)

Planarity Grid Drawings of Four-Connected Plane Graphs . . . . . . . . . . . . . . . . . . . . . . . 145 Kazuyuki Miura, Takao Nishizeki (Tohoku University) and Shin-ichi Nakano (Gunma University) Graph Embedding with Topological Cycle-Constraints . . . . . . . . . . . . . . . . . 155 Christoph Dornheim (University of Freiburg) Embedding Vertices at Points: Few Bends Suffice for Planar Graphs . . . . . . 165 Michael Kaufmann, Roland Wiese (Universit¨ at T¨ ubingen) The Constrained Crossing Minimization Problem . . . . . . . . . . . . . . . . . . . . . . 175 Petra Mutzel, Thomas Ziegler (Max-Planck-Institut f¨ ur Informatik)

Clusters II Planarity-Preserving Clustering and Embedding for Large Planar Graphs 186 Christian A. Duncan, Michael T. Goodrich and Stephen G. Kobourov (The John Hopkins University) An Algorithm for Drawing Compound Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 197 Fran¸cois Bertault, Mirka Miller (University of Newcastle)

Table of Contents

XI

Levels II The Vertex-Exchange Graph: A New Concept for Multi-level Crossing Minimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Patrick Healy, Ago Kuusik (University of Limerick) Using Sifting for k-Layer Straightline Crossing Minimization . . . . . . . . . . . 217 Christian Matuszewski, Robby Sch¨ onfeld, and Paul Molitor (University Halle-Wittenberg) On 3-Layer Crossings and Pseudo Arrangements . . . . . . . . . . . . . . . . . . . . . . . 225 Farhad Shahrokhi (University of North Texas), Imrich Vrt’o (Slovak Academy of Sciences)

Applications Visualizing Algorithms for the Design and Analysis of Survivable Networks 232 Ala Eddine Barouni (University of Tunis), Ali Jaoua (University Dhahran) and Nejib Zaguia (Ottawa) LayoutShow: A Signed Applet/Application for Graph Drawing and Experimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 Lila Behzadi (York University) Centrality in Policy Network Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Ulrik Brandes, Dorothea Wagner (University of Konstanz) and Patrick Kenis (Free University, Amsterdam ) Straight-Line Drawings of Protein Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 259 Wojciech Basalaj (University of Cambridge Computer Laboratory), Karen Eilbeck (University of Manchester Biochemistry Division)

Invited Talk Art of Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Jaroslav Neˇsetˇril (Charles University)

Symmetry An Heuristic for Graph Symmetry Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Hubert de Fraysseix (CRNS UMR, Paris) Isomorphic Subgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 Sabine Bachl (University of Passau)

Orthogonality II Orthogonal and Quasi-upward Drawings with Vertices of Prescribed Size . . 297 G. Di Battista, W. Didimo, M. Patrignani and M. Pizzonia (Universit` a di Roma Tre)

XII

Table of Contents

Multi-dimensional Orthogonal Graph Drawing with Small Boxes . . . . . . . . . 311 David R. Wood (Monash University)

Representations Geometric Realization of Simplicial Complexes . . . . . . . . . . . . . . . . . . . . . . . . 323 Patrice Ossona de Mendez (CNRS UMR, Paris) Visibility Representations of Complete Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 333 Robert Babilon, Helena Nyklov´ a , Ondˇrej Pangr´ ac and Jan Vondr´ ak (Charles University) Triangle-Free Planar Graphs as Segments Intersection Graphs . . . . . . . . . . . 341 N. de Castro, F. J. Cobos, J. C. Dana, A. M´ arquez (Universidad de Sevilla) and Marc Noy (Universitat Polit`ecnica de Catalunya)

Drawing II A Force-Directed Algorithm that Preserves Edge Crossing Properties . . . . . 351 Fran¸cois Bertault (University of Newcastle)

Proximity and Trees Rectangle of Influence Drawings of Graphs without Filled 3-Cycles . . . . . . 359 Therese Biedl (University of Waterloo), Anna Bretscher (Queen’s University) and Henk Meijer (Queen’s University) Voronoi Drawings of Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 Giuseppe Liotta (Univ. of Perugia), Henk Meijer (Queen’s Univ.) Infinite Trees and the Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 C. Demetrescu, Irene Finocchi (Universit` a di Roma “La Sapienza”), Giuseppe Di Battista, Maurizio Patrignani, Maurizio Pizzonia (Universit` a di Roma Tre) and Giuseppe Liotta (Universit` a di Perugia), Latour — A Tree Visualisation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 Ivan Herman, Guy Melan¸con, Maurice M de Ruiter (Centrum voor Wiskunde en Informatica) and Maylis Delest (Universit´e Bordeaux)

Graph Drawing Contest Graph-Drawing Contest Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 Franz J. Brandenburg, Falk Schreiber (Universit¨ at Passau), Michael J¨ unger (Universit¨ at zu K¨ oln), Joe Marks (MERL) and Petra Mutzel (Max-Planck-Institut f¨ ur Informatik) Hunting Down Graph B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 Ulrik Brandes (Brown University)

Table of Contents

XIII

Posters Orthogonal and Straight-Line Drawings of Graphs with Succinct Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 Ho-Lin Chen, Hsu-Chun Yen (National Taiwan University) Electronic Biochemical Pathways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 Carl-Christian Kanne (Universit¨ at Mannheim), Falk Schreiber (Universit¨ at Passau) and Dietrich Tr¨ umbach (Universit¨ at Erlangen-N¨ urnberg)

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

The Anatomy of a Geometric Algorithm? Jiˇr´ı Matouˇsek Department of Applied Mathematics, Charles University Prague, Czech Republic [email protected]

?

This article contains the speaker’s slides for the talk, reproduced by the editors with speaker’s permission. An appendix with references was prepared by the speaker.

J. Kratochv´ıl (Ed.): GD’99, LNCS 1731, pp. 1–7, 1999. c Springer-Verlag Berlin Heidelberg 1999

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