Universal Point Sets for Planar Graph Drawings with Circular Arcs
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Drawing Graphs and Maps with Curves
Report from Dagstuhl Seminar 13151 Drawing Graphs and Maps with Curves Edited by Stephen Kobourov1, Martin Nöllenburg2, and Monique Teillaud3 1 University of Arizona – Tucson, US, [email protected] 2 KIT – Karlsruhe Institute of Technology, DE, [email protected] 3 INRIA Sophia Antipolis – Méditerranée, FR, [email protected] Abstract This report documents the program and the outcomes of Dagstuhl Seminar 13151 “Drawing Graphs and Maps with Curves”. The seminar brought together 34 researchers from different areas such as graph drawing, information visualization, computational geometry, and cartography. During the seminar we started with seven overview talks on the use of curves in the different communities represented in the seminar. Abstracts of these talks are collected in this report. Six working groups formed around open research problems related to the seminar topic and we report about their findings. Finally, the seminar was accompanied by the art exhibition Bending Reality: Where Arc and Science Meet with 40 exhibits contributed by the seminar participants. Seminar 07.–12. April, 2013 – www.dagstuhl.de/13151 1998 ACM Subject Classification I.3.5 Computational Geometry and Object Modeling, G.2.2 Graph Theory, F.2.2 Nonnumerical Algorithms and Problems Keywords and phrases graph drawing, information visualization, computational cartography, computational geometry Digital Object Identifier 10.4230/DagRep.3.4.34 Edited in cooperation with Benjamin Niedermann 1 Executive Summary Stephen Kobourov Martin Nöllenburg Monique Teillaud License Creative Commons BY 3.0 Unported license © Stephen Kobourov, Martin Nöllenburg, and Monique Teillaud Graphs and networks, maps and schematic map representations are frequently used in many fields of science, humanities and the arts. -
Linear Size Universal Point Sets for Classes of Planar Graphs⋆
Lines: 400 Linear size universal point sets for classes of planar graphs? Stefan Felsner[0000−0002−6150−1998], Hendrik Schrezenmaier[0000−0002−1671−9314], Felix Schr¨oder[0000−0001−8563−3517], and Raphael Steiner[0000−0002−4234−6136] Institute of Mathematics, Technische Universit¨atBerlin, Germany ffelsner,fschroed,schrezen,[email protected] 2 1 Abstract. A finite point set P ⊆ R is n-universal with respect to a 2 class G if every n-vertex-graph G 2 G admits a crossing-free straight-line 3 drawing with vertices being placed at points of P . A popular task in 4 graph drawing is to identify small universal point sets. 5 For the class of all planar graphs the best known upper bound on the 6 size of a universal point set is quadratic and the best known lower bound 7 is linear in n. One of the classical results in the area is that every set 8 of n points in general position (no three collinear) is n-universal for out- 9 erplanar graphs. While some other classes are known to admit universal 10 point sets of near linear size, we are not aware of truly linear bounds for 11 interesting classes beyond outerplanar graphs. 12 In this paper we study a specific ordered point set H (the exploding 13 double chain) and show that all planar graphs G on n ≥ 2 vertices which 14 are subgraphs of a planar graph admitting a one-sided Hamiltonian cycle 15 have a straight-line drawing on the initial piece Hn of size 2n − 2 in H. -
Practical Course on Graph Visualization Introduction
Practical Course on Graph Visualization Introduction SUMMER SEMESTER 2014/2015 Tamara Mchedlidze – Martin Nollenburg¨ VS KIT – University of the State of Baden-Wuerttemberg and Institute of Theoreticalwww.kit.edu Informatics TamaraNational Mchedlidze Laboratory – of Practical the Helmholtz Course Association on Graph Visualization Prof. Dr. Dorothea Wagner Organization Instructors Tamara Mchedlidze [email protected] Office 307 Martin Nollenburg¨ [email protected] Office 319 Class meetings Tuesday 11:30 - 13:00 Room SR301 Detailed plan of meetings on the webpage Institute of Theoretical Informatics Tamara Mchedlidze – Practical Course on Graph Visualization Prof. Dr. Dorothea Wagner Organization Web-page http://i11www.iti.uni-karlsruhe.de/teaching/sommer2015/graphvis/ Information on the web-page Meeting dates (subject to change) Deadlines and goals Literature and reading material (...) Institute of Theoretical Informatics Tamara Mchedlidze – Practical Course on Graph Visualization Prof. Dr. Dorothea Wagner Practical Course on Graph Visualization Winter Semester Summer Semester Algorithms for Practical Course on Graph Visualization Graph Visualization Institute of Theoretical Informatics Tamara Mchedlidze – Practical Course on Graph Visualization Prof. Dr. Dorothea Wagner Practical Course on Graph Visualization Winter Semester Summer Semester Algorithms for Practical Course on Graph Visualization Graph Visualization Practical course is part of the module Theory and Practice of Graph Visualization (IN4INGTP) Module consists of the theory lecture (5 credits) and the practical course (5 credits) Institute of Theoretical Informatics Tamara Mchedlidze – Practical Course on Graph Visualization Prof. Dr. Dorothea Wagner Practical Course on Graph Visualization Winter Semester Summer Semester Algorithms for Practical Course on Graph Visualization Graph Visualization Institute of Theoretical Informatics Tamara Mchedlidze – Practical Course on Graph Visualization Prof. -
A Survey on Graph Drawing Beyond Planarity
A Survey on Graph Drawing Beyond Planarity Walter Didimo1, Giuseppe Liotta1, Fabrizio Montecchiani1 1Dipartimento di Ingegneria, Universit`adegli Studi di Perugia, Italy fwalter.didimo,giuseppe.liotta,[email protected] Abstract Graph Drawing Beyond Planarity is a rapidly growing research area that classifies and studies geometric representations of non-planar graphs in terms of forbidden crossing configurations. Aim of this survey is to describe the main research directions in this area, the most prominent known results, and some of the most challenging open problems. 1 Introduction In the mid 1980s, the early pioneers of graph drawing had the intuition that a drawing with too many edge crossings is harder to read than a drawing of the same graph with fewer edge crossings (see, e.g., [35, 36, 72, 190]). This intuition was later confirmed by a series of cognitive experimental studies (see, e.g., [180, 181, 203]). As a result, a large part of the existing literature on graph drawing showcases elegant algorithms and sophisticated data structures under the assumption that the input graph is planar, i.e., it admits a drawing without edge crossings. When the input graph is non-planar, crossing minimization heuristics are used to insert a small number of dummy vertices in correspondence of the edge crossings, so to obtain a planarization of the input graph. A crossing-free drawing of the planarization can be computed by using one of the algorithms for planar graphs and then the crossings are reinserted by removing the dummy vertices. This approach is commonly adopted and works well for graphs of relatively small size, up to a few hundred vertices and edges (see, e.g., [87, 152]). -
Superpatterns and Universal Point Sets Michael J
Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 0, no. 0, pp. 0{0 (0) DOI: 10.7155/jgaa.00318 Superpatterns and Universal Point Sets Michael J. Bannister Zhanpeng Cheng William E. Devanny David Eppstein Department of Computer Science, University of California, Irvine Abstract An old open problem in graph drawing asks for the size of a universal point set, a set of points that can be used as vertices for straight-line draw- ings of all n-vertex planar graphs. We connect this problem to the theory of permutation patterns, where another open problem concerns the size of superpatterns, permutations that contain all patterns of a given size. We generalize superpatterns to classes of permutations determined by for- bidden patterns, and we construct superpatterns of size n2=4 + Θ(n) for the 213-avoiding permutations, half the size of known superpatterns for unconstrained permutations. We use our superpatterns to construct uni- versal point sets of size n2=4−Θ(n), smaller than the previous bound by a 9/16 factor. We prove that every proper subclass of the 213-avoiding per- mutations has superpatterns of size O(n logO(1) n), which we use to prove that the planar graphs of bounded pathwidth have near-linear universal point sets. Submitted: Reviewed: Revised: Accepted: Final: December 2013 February 2014 March 2014 March 2014 April 2014 Published: Article type: Communicated by: Regular Paper S. Wismath and A. Wolff This research was supported in part by the National Science Foundation under grants 0830403 and 1217322, and by the Office of Naval Research under MURI grant N00014-08-1-1015. -
Crossing Minimization for 1-Page and 2-Page Drawings of Graphs with Bounded Treewidth
Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 22, no. 4, pp. 577{606 (2018) DOI: 10.7155/jgaa.00479 Crossing Minimization for 1-page and 2-page Drawings of Graphs with Bounded Treewidth Michael J. Bannister 1 David Eppstein 2 1Pinterest, San Francisco, California, USA 2Dept. of Computer Science, University of California, Irvine, USA Abstract We investigate crossing minimization for 1-page and 2-page book drawings. We show that computing the 1-page crossing number is fixed- parameter tractable with respect to the number of crossings, that testing 2-page planarity is fixed-parameter tractable with respect to treewidth, and that computing the 2-page crossing number is fixed-parameter tractable with respect to the sum of the number of crossings and the treewidth of the input graph. We prove these results via Courcelle's theorem on the fixed-parameter tractability of properties expressible in monadic second order logic for graphs of bounded treewidth. Submitted: Reviewed: Revised: Accepted: Final: March 2018 August 2018 September 2018 October 2018 October 2018 Published: December 2018 Article type: Communicated by: Regular paper W.S. Evans This material is based upon work supported by the National Science Foundation under Grants CCF-1228639, CCF-1618301, and CCF-1616248, and by the Office of Naval Research under Grant No. N00014-08-1-1015. A preliminary version of this work appeared in the Proceedings of the 22nd International Symposium on Graph Drawing (GD 2014) [6]. E-mail addresses: [email protected] (Michael J. Bannister) [email protected] (David Eppstein) 578 Bannister and Eppstein Crossing Minimization for 1-page and 2-page.. -
Crossing Minimization for 1-Page and 2-Page Drawings of Graphs with Bounded Treewidth
Crossing Minimization for 1-page and 2-page Drawings of Graphs with Bounded Treewidth Michael J. Bannister and David Eppstein Department of Computer Science, University of California, Irvine Abstract. We investigate crossing minimization for 1-page and 2-page book drawings. We show that computing the 1-page crossing number is fixed-parameter tractable with respect to the number of crossings, that testing 2-page planarity is fixed-parameter tractable with respect to treewidth, and that computing the 2-page crossing number is fixed- parameter tractable with respect to the sum of the number of crossings and the treewidth of the input graph. We prove these results via Cour- celle’s theorem on the fixed-parameter tractability of properties express- ible in monadic second order logic for graphs of bounded treewidth. 1 Introduction A k-page book embedding of a graph G is a drawing that places the vertices of G on a line (the spine of the book) and draws each edge, without crossings, inside one of k half-planes bounded by the line (the pages of the book) [16,19]. In one common drawing style, an arc diagram, the edges in each page are drawn as circular arcs perpendicular to the spine [24], but the exact shape of the edges is unimportant for the existence of book embeddings. These embeddings can be generalized to k-pagebookdrawings: as before, we place each vertex on the spine and each edge within a single page, but with crossings allowed. The crossing number of such a drawing is defined to be the sum of the numbers of crossings within each page, and the k-page crossing number crk(G) is the minimum number of crossings in any k-page book drawing [22]. -
Arc Diagrams, Flip Distances, and Hamiltonian Triangulations
Research Collection Conference Paper Arc Diagrams, Flip Distances, and Hamiltonian Triangulations Author(s): Cardinal, Jean; Hoffmann, Michael; Kusters, Vincent; Tóth, Csaba D.; Wettstein, Manuel Publication Date: 2015 Permanent Link: https://doi.org/10.3929/ethz-b-000109753 Originally published in: Leibniz International Proceedings in Informatics 30, http://doi.org/10.4230/LIPIcs.STACS.2015.197 Rights / License: Creative Commons Attribution 3.0 Unported This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library Arc Diagrams, Flip Distances, and Hamiltonian Triangulations Jean Cardinal∗1, Michael Hoffmann†2, Vincent Kusters†2, Csaba D. Tóth3, and Manuel Wettstein‡2 1 Université libre de Bruxelles (ULB), Belgium [email protected] 2 Department of Computer Science, ETH Zürich, Switzerland {hoffmann,vincent.kusters,manuelwe}@inf.ethz.ch 3 California State University Northridge, Los Angeles, CA, USA [email protected] Abstract We show that every triangulation (maximal planar graph) on n ≥ 6 vertices can be flipped into a Hamiltonian triangulation using a sequence of less than n/2 combinatorial edge flips. The previously best upper bound uses 4-connectivity as a means to establish Hamiltonicity. But in general about 3n/5 flips are necessary to reach a 4-connected triangulation. Our result improves the upper bound on the diameter of the flip graph of combinatorial triangulations on n vertices from 5.2n − 33.6 to 5n − 23. We also show that for every triangulation on n vertices there is a simultaneous flip of less than 2n/3 edges to a 4-connected triangulation. -
Selected Open Problems in Graph Drawing
Selected Open Problems in Graph Drawing Franz Brandenburg?, David Eppstein??, Michael T. Goodrich???, Stephen Kobourovy, Giuseppe Liottaz, and Petra Mutzelx Abstract. In this manuscript, we present several challenging and inter- esting open problems in graph drawing. The goal of the listing in this paper is to stimulate future research in graph drawing. 1 Introduction This paper is devoted to exploring some interesting and challenging open prob- lems in graph drawing. We have specifically chosen topics motivated from re- search themes of current interest in graph drawing, including proximity draw- ings, 2D and 3D straight-line drawings, 2D and 3D orthogonal representations, and graph drawing checkers. Our choice of open problems and themes should not be seen as exhaustive, however, as graph drawing is a rich area of research with many open problems and research themes. For example, there are interesting open problems related to the routing of multiple curves in the plane, dynamic graph drawing, drawing of hypergraphs and Venn diagrams, applied graph draw- ing, and label placement in graph drawings, which we do not specifically address in this paper. In any survey of open problems, there is naturally going to be a certain em- phasis that reflects the perspective of the authors, and this paper is no exception. Nevertheless, we have striven to include open problems motivated by topics we believe are of general interest. That is, we feel that the solution to any of the open problems we include in this paper would be of wide interest in the graph drawing community. 2 Proximity Drawability Problems Recently, much attention has been devoted to the study of the combinatorial properties of different types of proximity graphs. -
Universal Point Sets for Drawing Planar Graphs with Circular Arcs
Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 18, no. 3, pp. 313{324 (2014) DOI: 10.7155/jgaa.00324 Universal Point Sets for Drawing Planar Graphs with Circular Arcs Patrizio Angelini 1 David Eppstein 2 Fabrizio Frati 3 Michael Kaufmann 4 Sylvain Lazard 5 Tamara Mchedlidze 6 Monique Teillaud 7 Alexander Wolff 8 1Dipartimento di Ingegneria, Roma Tre University, Italy 2Computer Science Department, University of California, Irvine, U.S.A. 3School of Information Technology, The University of Sydney, Australia 4Wilhelm-Schickard-Institut f¨urInformatik, Universit¨atT¨ubingen,Germany 5INRIA Nancy Grand Est { Loria, France 6Institute of Theor. Informatics, Karlsruhe Institute of Technology, Germany 7INRIA Sophia Antipolis { M´editerran´ee,France 8Lehrstuhl f¨urInformatik I, Universit¨atW¨urzburg,Germany Abstract We prove that there exists a set S of n points in the plane such that every n-vertex planar graph G admits a planar drawing in which every vertex of G is placed on a distinct point of S and every edge of G is drawn as a circular arc. Submitted: Reviewed: Revised: Accepted: Final: October 2013 February 2014 March 2014 April 2014 May 2014 Published: May 2014 Article type: Communicated by: Regular paper Csaba D. T´oth An earlier version of this paper appeared as \Universal Point Sets for Planar Graph Drawings with Circular Arcs", Proc. 25th Canadian Conf. Comput. Geom., pages 117{122, 2013. David Eppstein was supported in part by the National Science Foundation under grants 0830403 and 1217322, and by the Office of Naval Research under MURI grant N00014-08-1- 1015. Patrizio Angelini, Michael Kaufmann, and Alexander Wolff acknowledge support by the ESF EuroGIGA project GraDR (DFG grants Ka 812/16-1 and Wo 758/5-1). -
Chronological Index
Chronological Index 1952; Shannon 548 1974; Dijkstra 812 Mobile Agents and Exploration Self-Stabilization 1955; Kuhn 68 1974; Elias 748 Assignment Problem Rank and Select Operations on Binary Strings 1956; McCluskey 989 1975; Ibarra, Kim 419 Two-Level Boolean Minimization Knapsack 1957; Munkres 68 1976; Booth, Lueker 656 Assignment Problem Planarity Testing 1959; Rosenblatt 642 Perceptron Algorithm 1976; Christofides 517 Metric TSP 1962; Gale, Shapley 390 Hospitals/Residents Problem 1977; Ziv, Lempel 236 Dictionary-Based Data Compression 1962; Gale, Shapley 877 Stable Marriage 1978; Lamport 129 Causal Order, Logical Clocks, State Machine 1965; Dijkstra 188 Replication Concurrent Programming, Mutual Exclusion 1980; McKay 373 1966; Graham 455 Graph Isomorphism List Scheduling 1980; Pease, Shostak, Lamport 116 1968; Coffman, Kleinrock 562 Byzantine Agreement Multi-level Feedback Queues 1972; Bayer, McCreight 108 1981; Kierstead, Trotter 594 B-trees Online Interval Coloring 1973; Liu, Layland 751 1982; Karmarker, Karp 57 Rate-Monotonic Scheduling Approximation Schemes for Bin Packing 1974–1979, Chvátal, Johnson, Lovász, Stein 379 1982; Lenstra, Lenstra, Lovasz 841 Greedy Set-Cover Algorithms Shortest Vector Problem 1040 Chronological Index 1983; Baker 59 1986; Lamport, Vitanyi, Awerbuch 761 Approximation Schemes for Planar Graph Problems Registers 1983; Case, Smith 411 1987; Arnborg, Corneil, Proskurowski 968 Inductive Inference Treewidth of Graphs 1983; Gallager, Humblet, Spira 256 1987; Irving, Leather, Gusfield -
Superpatterns and Universal Point Sets Michael J
Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 18, no. 2, pp. 177{209 (2014) DOI: 10.7155/jgaa.00318 Superpatterns and Universal Point Sets Michael J. Bannister Zhanpeng Cheng William E. Devanny David Eppstein Department of Computer Science, University of California, Irvine Abstract An old open problem in graph drawing asks for the size of a universal point set, a set of points that can be used as vertices for straight-line draw- ings of all n-vertex planar graphs. We connect this problem to the theory of permutation patterns, where another open problem concerns the size of superpatterns, permutations that contain all patterns of a given size. We generalize superpatterns to classes of permutations determined by for- bidden patterns, and we construct superpatterns of size n2=4 + Θ(n) for the 213-avoiding permutations, half the size of known superpatterns for unconstrained permutations. We use our superpatterns to construct uni- versal point sets of size n2=4−Θ(n), smaller than the previous bound by a 9/16 factor. We prove that every proper subclass of the 213-avoiding per- mutations has superpatterns of size O(n logO(1) n), which we use to prove that the planar graphs of bounded pathwidth have near-linear universal point sets. Submitted: Reviewed: Revised: Accepted: Final: December 2013 February 2014 March 2014 March 2014 April 2014 Published: May 2014 Article type: Communicated by: Regular Paper S. Wismath and A. Wolff This research was supported in part by the National Science Foundation under grants 0830403 and 1217322, and by the Office of Naval Research under MURI grant N00014-08-1-1015.