Representing Graphs by Polygons with Edge Contacts in 3D
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Equivelar Octahedron of Genus 3 in 3-Space
Equivelar octahedron of genus 3 in 3-space Ruslan Mizhaev ([email protected]) Apr. 2020 ABSTRACT . Building up a toroidal polyhedron of genus 3, consisting of 8 nine-sided faces, is given. From the point of view of topology, a polyhedron can be considered as an embedding of a cubic graph with 24 vertices and 36 edges in a surface of genus 3. This polyhedron is a contender for the maximal genus among octahedrons in 3-space. 1. Introduction This solution can be attributed to the problem of determining the maximal genus of polyhedra with the number of faces - . As is known, at least 7 faces are required for a polyhedron of genus = 1 . For cases ≥ 8 , there are currently few examples. If all faces of the toroidal polyhedron are – gons and all vertices are q-valence ( ≥ 3), such polyhedral are called either locally regular or simply equivelar [1]. The characteristics of polyhedra are abbreviated as , ; [1]. 2. Polyhedron {9, 3; 3} V1. The paper considers building up a polyhedron 9, 3; 3 in 3-space. The faces of a polyhedron are non- convex flat 9-gons without self-intersections. The polyhedron is symmetric when rotated through 1804 around the axis (Fig. 3). One of the features of this polyhedron is that any face has two pairs with which it borders two edges. The polyhedron also has a ratio of angles and faces - = − 1. To describe polyhedra with similar characteristics ( = − 1) we use the Euler formula − + = = 2 − 2, where is the Euler characteristic. Since = 3, the equality 3 = 2 holds true. -
The Circle Packing Theorem
Alma Mater Studiorum · Università di Bologna SCUOLA DI SCIENZE Corso di Laurea in Matematica THE CIRCLE PACKING THEOREM Tesi di Laurea in Analisi Relatore: Pesentata da: Chiar.mo Prof. Georgian Sarghi Nicola Arcozzi Sessione Unica Anno Accademico 2018/2019 Introduction The study of tangent circles has a rich history that dates back to antiquity. Already in the third century BC, Apollonius of Perga, in his exstensive study of conics, introduced problems concerning tangency. A famous result attributed to Apollonius is the following. Theorem 0.1 (Apollonius - 250 BC). Given three mutually tangent circles C1, C2, 1 C3 with disjoint interiors , there are precisely two circles tangent to all the three initial circles (see Figure1). A simple proof of this fact can be found here [Sar11] and employs the use of Möbius transformations. The topic of circle packings as presented here, is sur- prisingly recent and originates from William Thurston's famous lecture notes on 3-manifolds [Thu78] in which he proves the theorem now known as the Koebe-Andreev- Thurston Theorem or Circle Packing Theorem. He proves it as a consequence of previous work of E. M. Figure 1 Andreev and establishes uniqueness from Mostov's rigid- ity theorem, an imporant result in Hyperbolic Geometry. A few years later Reiner Kuhnau pointed out a 1936 proof by german mathematician Paul Koebe. 1We dene the interior of a circle to be one of the connected components of its complement (see the colored regions in Figure1 as an example). i ii A circle packing is a nite set of circles in the plane, or equivalently in the Riemann sphere, with disjoint interiors and whose union is connected. -
Analytical Framework for Recurrence-Network Analysis of Time Series
Analytical framework for recurrence-network analysis of time series Jonathan F. Donges,1, 2, ∗ Jobst Heitzig,1 Reik V. Donner,1 and J¨urgenKurths1, 2, 3 1Potsdam Institute for Climate Impact Research, P.O. Box 601203, 14412 Potsdam, Germany 2Department of Physics, Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany 3Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3FX, United Kingdom (Dated: August 13, 2018) Recurrence networks are a powerful nonlinear tool for time series analysis of complex dynamical systems. While there are already many successful applications ranging from medicine to paleoclima- tology, a solid theoretical foundation of the method has still been missing so far. Here, we interpret an "-recurrence network as a discrete subnetwork of a \continuous" graph with uncountably many vertices and edges corresponding to the system's attractor. This step allows us to show that various statistical measures commonly used in complex network analysis can be seen as discrete estimators of newly defined continuous measures of certain complex geometric properties of the attractor on the scale given by ". In particular, we introduce local measures such as the "-clustering coefficient, mesoscopic measures such as "-motif density, path-based measures such as "-betweennesses, and global measures such as "-efficiency. This new analytical basis for the so far heuristically motivated network measures also provides an objective criterion for the choice of " via a percolation threshold, and it shows that estimation can be improved by so-called node splitting invariant versions of the measures. We finally illustrate the framework for a number of archetypical chaotic attractors such as those of the Bernoulli and logistic maps, periodic and two-dimensional quasi-periodic motions, and for hyperballs and hypercubes, by deriving analytical expressions for the novel measures and com- paring them with data from numerical experiments. -
Colouring Contact Graphs of Squares and Rectilinear Polygons
Colouring contact graphs of squares and rectilinear polygons Citation for published version (APA): de Berg, M., Markovic, A., & Woeginger, G. (2016). Colouring contact graphs of squares and rectilinear polygons. In 32nd European Workshop on Computational Geometry (EuroCG 2016), 30 March - 1 April, Lugano, Switzerland (pp. 71-74) Document status and date: Published: 01/01/2016 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. -
Single Digits
...................................single digits ...................................single digits In Praise of Small Numbers MARC CHAMBERLAND Princeton University Press Princeton & Oxford Copyright c 2015 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved The second epigraph by Paul McCartney on page 111 is taken from The Beatles and is reproduced with permission of Curtis Brown Group Ltd., London on behalf of The Beneficiaries of the Estate of Hunter Davies. Copyright c Hunter Davies 2009. The epigraph on page 170 is taken from Harry Potter and the Half Blood Prince:Copyrightc J.K. Rowling 2005 The epigraphs on page 205 are reprinted wiht the permission of the Free Press, a Division of Simon & Schuster, Inc., from Born on a Blue Day: Inside the Extraordinary Mind of an Austistic Savant by Daniel Tammet. Copyright c 2006 by Daniel Tammet. Originally published in Great Britain in 2006 by Hodder & Stoughton. All rights reserved. Library of Congress Cataloging-in-Publication Data Chamberland, Marc, 1964– Single digits : in praise of small numbers / Marc Chamberland. pages cm Includes bibliographical references and index. ISBN 978-0-691-16114-3 (hardcover : alk. paper) 1. Mathematical analysis. 2. Sequences (Mathematics) 3. Combinatorial analysis. 4. Mathematics–Miscellanea. I. Title. QA300.C4412 2015 510—dc23 2014047680 British Library -
Geometric Biplane Graphs I: Maximal Graphs ∗
Geometric Biplane Graphs I: Maximal Graphs ∗ Alfredo Garc´ıa1 Ferran Hurtado2 Matias Korman3;4 In^esMatos5 Maria Saumell6 Rodrigo I. Silveira5;2 Javier Tejel1 Csaba D. T´oth7 Abstract We study biplane graphs drawn on a finite planar point set S in general position. This is the family of geometric graphs whose vertex set is S and can be decomposed into two plane graphs. We show that two maximal biplane graphs|in the sense that no edge can be added while staying biplane|may differ in the number of edges, and we provide an efficient algorithm for adding edges to a biplane graph to make it maximal. We also study extremal properties of maximal biplane graphs such as the maximum number of edges and the largest maximum connectivity over n-element point sets. 1 Introduction In a geometric graph G = (V; E), the vertices are distinct points in the plane in general position, and the edges are straight line segments between pairs of vertices. A plane graph is a geometric graph in which no two edges cross. Every (abstract) graph has a realization as a geometric graph (by simply mapping the vertices into distinct points in the plane, no three of which are collinear), and every planar graph can be realized as a plane graph by F´ary'stheorem [13]. The number of n-vertex labeled planar graphs is at least 27:22n n! [15]. · However, there are only 2O(n) plane graphs on any given set of n points in the plane [2, 22]. We consider a generalization of plane graphs. -
Penrose M. Random Geometric Graphs (OUP, 2003)(ISBN 0198506260)(345S) Mac .Pdf
OXFORD STUDIES IN PROBABILITYManaging Editor L. C. G. ROGERS Editorial Board P. BAXENDALE P. GREENWOOD F. P. KELLY J.-F. LE GALL E. PARDOUX D. WILLIAMS OXFORD STUDIES IN PROBABILITY 1. F. B. Knight: Foundations of the prediction process 2. A. D. Barbour, L. Holst, and S. Janson: Poisson approximation 3. J. F. C. Kingman: Poisson processes 4. V. V. Petrov: Limit theorems of probability theory 5. M. Penrose: Random geometric graphs Random Geometric Graphs MATHEW PENROSE University of Bath Great Clarendon Street, Oxford OX26DP Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Kaarachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi São Paulo Shanghai Taipei Tokyo Toronto Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Mathew Penrose, 2003 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2003 Reprinted 2004 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organisation. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above. -
COLORING GRAPHS USING TOPOLOGY 11 Realized by Tutte Who Called an Example of a Disc G for Which the Boundary Has Chromatic Number 4 a Chromatic Obstacle
COLORING GRAPHS USING TOPOLOGY OLIVER KNILL Abstract. Higher dimensional graphs can be used to color 2- dimensional geometric graphs G. If G the boundary of a three dimensional graph H for example, we can refine the interior until it is colorable with 4 colors. The later goal is achieved if all inte- rior edge degrees are even. Using a refinement process which cuts the interior along surfaces S we can adapt the degrees along the boundary S. More efficient is a self-cobordism of G with itself with a host graph discretizing the product of G with an interval. It fol- lows from the fact that Euler curvature is zero everywhere for three dimensional geometric graphs, that the odd degree edge set O is a cycle and so a boundary if H is simply connected. A reduction to minimal coloring would imply the four color theorem. The method is expected to give a reason \why 4 colors suffice” and suggests that every two dimensional geometric graph of arbitrary degree and ori- entation can be colored by 5 colors: since the projective plane can not be a boundary of a 3-dimensional graph and because for higher genus surfaces, the interior H is not simply connected, we need in general to embed a surface into a 4-dimensional simply connected graph in order to color it. This explains the appearance of the chromatic number 5 for higher degree or non-orientable situations, a number we believe to be the upper limit. For every surface type, we construct examples with chromatic number 3; 4 or 5, where the construction of surfaces with chromatic number 5 is based on a method of Fisk. -
Geometric Intersection Patterns and the Theory of Topological Graphs
Geometric Intersection Patterns and the Theory of Topological Graphs J´anosPach∗ Abstract. The intersection graph of a set system S is a graph on the vertex set S, in which two vertices are connected by an edge if and only if the corresponding sets have nonempty intersection. It was shown by Tietze (1905) that every finite graph is the intersection graph of 3-dimensional convex polytopes. The analogous statement is false in any fixed dimension if the polytopes are allowed to have only a bounded number of faces or are replaced by simple geometric objects that can be described in terms of a bounded number of real parameters. Intersection graphs of various classes of geometric objects, even in the plane, have interesting structural and extremal properties. We survey problems and results on geometric intersection graphs and, more gener- ally, intersection patterns. Many of the questions discussed were originally raised by Berge, Erd}os,Gr¨unbaum, Hadwiger, Tur´an,and others in the context of classical topol- ogy, graph theory, and combinatorics (related, e.g., to Helly's theorem, Ramsey theory, perfect graphs). The rapid development of computational geometry and graph drawing algorithms in the last couple of decades gave further impetus to research in this field. A topological graph is a graph drawn in the plane so that its vertices are represented by points and its edges by possibly intersecting simple continuous curves connecting the corresponding point pairs. We give applications of the results concerning intersection patterns in the theory of topological graphs. Mathematics Subject Classification (2010). Primary 05C35; Secondary 05C62, 52C10. -
Compatible Geometric Matchings$
Compatible Geometric MatchingsI Oswin Aichholzera,1, Sergey Beregb, Adrian Dumitrescuc,2, Alfredo Garc´ıad,3, Clemens Huemere,4, Ferran Hurtadoe,4, Mikio Kanof, Alberto M´arquezg, David Rappaporth,5, Shakhar Smorodinskyi, Diane Souvainej, Jorge Urrutiak,6, David R. Woodl,∗,4,7 aInstitute for Software Technology, Graz University of Technology, Austria bDepartment of Computer Science, University of Texas at Dallas, U.S.A. cDepartment of Computer Science, University of Wisconsin-Milwaukee, U.S.A. dDepartamento de M´etodos Estad´ısticos, Universidad de Zaragoza, Spain eDepartament de Matem`atica Aplicada II, Universitat Polit`ecnica de Catalunya, Spain fDepartment of Computer and Information Sciences, Ibaraki University, Japan gDepartamento de Matem´atica Aplicada I, Universidad de Sevilla, Spain hSchool of Computing, Queen's University, Canada iDepartment of Mathematics, Ben-Gurion University, Israel jDepartment of Computer Science, Tufts University, U.S.A. kInstituto de Matem´aticas, Universidad Nacional Aut´onomade M´exico, M´exico lDepartment of Mathematics and Statistics, The University of Melbourne, Australia Abstract This paper studies non-crossing geometric perfect matchings. Two such per- fect matchings are compatible if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect match- ings M and M 0 of the same set of n points, for some k 2 O(log n), there 0 is a sequence of perfect matchings M = M0;M1;:::;Mk = M , such that each Mi is compatible with Mi+1. This improves the previous best bound of k ≤ n − 2. We then study the conjecture: every perfect matching with an even number of edges has an edge-disjoint compatible perfect matching. -
On Primal-Dual Circle Representations
On Primal-Dual Circle Representations Stefan Felsner∗ G¨unter Rote y The Koebe-Andreev-Thurston Circle Packing Theorem states that every triangulated planar graph has a circle-contact representation. The theorem has been generalized in various ways. The arguably most prominent generalization assures the existence of a primal-dual circle representation for every 3-connected planar graph. The aim of this note is to give a streamlined proof of this result. 1 Introduction For a 3-connected plane graph G = (V; E) with face set F , a primal-dual circle representation of G consists of two families of circles (Cx : x 2 V ) and (Dy : y 2 F ) such that: (i) The vertex-circles Cx have pairwise disjoint interiors. (ii) All face-circles Dy are contained in the circle Do corresponding to the outer face o, and all other face-circles have pairwise disjoint interiors. Moreover, for every edge xx0 2 E with dual edge yy0 (i. e., y and y0 are the two faces separated by xx0), the following holds: (iii) Circles Cx and Cx0 are tangent at a point p with tangent line txx0 . (iv) Circles Dy and Dy0 are tangent at the same point p with tangent line tyy0 . (v) The lines txx0 and tyy0 are orthogonal. Figure 1 shows an example. Theorem 1. Every 3-connected plane graph G admits a primal-dual circle representation. Moreover this representation is unique up to M¨obiustransformations. As a special case of this statement, G admits a circle packing representation: as a contact graph of nonoverlapping disks. The proof presented here combines ideas from an unpublished manuscript of Pulleyblank and Rote, from Brightwell and Scheinerman [5] and from Mohar [23]. -
A Note on Light Geometric Graphs
A note on light geometric graphs Eyal Ackerman∗ Jacob Foxy Rom Pinchasiz March 19, 2012 Abstract Let G be a geometric graph on n vertices in general position in the plane. We say that G is k-light if no edge e of G has the property that each of the two open half-planes bounded by the line through e contains more than k edges of G. We extend the previous result in [1] and withp a shorter argument show that every k-light geometric graph on n vertices has at most O(n k) edges. This bound is best possible. Keywords: Geometric graphs, k-near bipartite. 1 Introduction Let G be an n-vertex geometric graph. That is, G is a graph drawn in the plane such that its vertices are distinct points and its edges are straight-line segments connecting corresponding vertices. It is usually assumed, as we will assume in this paper, that the set of vertices of G is in general position in the sense that no three of them lie on a line. A typical question in geometric graph theory asks for the maximum number of edges that a geometric graph on n vertices can have assuming a forbidden configuration in that graph. This is a popular area of study extending classical extremal graph theory, utilizing diverse tools from both geometry and combinatorics. For example, an old result of Hopf and Pannwitz [3] and independently Sutherland [7] states that any geometric graph on n vertices with no pair of disjoint edges has at most n edges.