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Dos Problemas De Combinatoria Geométrica: Triangulaciones

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Dos Problemas De Combinatoria Geométrica: Triangulaciones Dos problemas de combinatoria geometrica:´ Triangulaciones eficientes del hipercubo; Grafos planos y rigidez. (Two problems in geometric combinatorics: Efficient triangulations of the hypercube; Planar graphs and rigidity). David Orden Mart´ın Universidad de Cantabria Tesis doctoral dirigida por el Profesor Francisco Santos Leal. Santander, Marzo de 2003. 2 ´Indice Agradecimientos iii Pre´ambulo v Preamble vii 1 Introducci´on y Preliminares 1 1.1 Triangulaciones y subdivisiones de politopos . 1 1.1.1 Preliminares sobre Teor´ıa de Politopos . 2 1.1.2 Triangulaciones de cubos . 6 1.2 Teor´ıa de Grafos . 11 1.2.1 Preliminares y relaciones con politopos . 11 1.2.2 Inmersiones planas de grafos: Teorema de Tutte . 15 1.3 Teor´ıa de Rigidez . 20 1.3.1 La matriz de rigidez . 20 1.3.2 Tensiones y grafos rec´ıprocos . 23 1.3.3 Rigidez infinitesimal y grafos isost´aticos . 26 1.4 Pseudo-triangulaciones de cuerpos convexos y puntos . 29 1.4.1 Pseudo-triangulaciones de cuerpos convexos . 29 1.4.2 Pseudo-triangulaciones de puntos . 31 1.5Resultados y Problemas abiertos . 35 1.5.1 Triangulaciones eficientes de cubos . 35 1.5.2 Grafos sin cruces, rigidez y pseudo-triangulaciones . 36 1 Introduction and Preliminaries 41 1.1 Triangulations and subdivisions of polytopes . 41 1.1.1 Preliminaries about Polytope Theory . 42 1.1.2 Triangulations of cubes . 46 1.2 Graph Theory . 50 1.2.1 Preliminaries and relations with polytopes . 51 1.2.2 Plane embeddings of graphs: Tutte’s Theorem . 54 1.3 Rigidity Theory . 59 1.3.1 The rigidity matrix . 59 1.3.2 Stresses and reciprocal graphs . 62 i ii 1.3.3 Infinitesimal rigidity and isostatic graphs . 65 1.4 Pseudo-triangulations of convex bodies and points . 67 1.4.1 Pseudo-triangulations of convex bodies . 68 1.4.2 Pseudo-triangulations of points . 69 1.5Results and Open problems . 73 1.5.1 Efficient triangulations of cubes . 73 1.5.2 Non-crossing graphs, rigidity and pseudo-triangulations . 74 2 Asymptotically efficient triangulations of the d-cube 79 2.1 Introduction . 79 2.2 Overview of the method and results . 82 ··· − − 2.3 Polyhedral subdivision of P ×∆k1+ +km 1 induced by one of P ×∆m 1 84 ··· − − 2.4 Triangulation of P × ∆k1+ +km 1 induced by one of P × ∆m 1 ... 86 2.5A triangulation of P × Q ........................ 88 2.6 Interpretation of our method via the Cayley Trick . 92 2.7 Computation of ρl,m ........................... 95 2.7.1 Smallest weighted efficiency of triangulations of I2 × ∆m−1 .95 2.7.2 Smallest weighted efficiency of triangulations of I3 × ∆m−1, for m =2, 3............................ 97 3 The polytope of non-crossing graphs on a planar point set 101 3.1 Introduction . 101 3.2 The graph of all pseudo-triangulations of A .............. 105 3.2.1 Marked non-crossing graphs on A. ............... 109 3.3 The polyhedron of marked non-crossing graphs on A ......... 110 3.4 Valid choices of f ............................. 116 3.5Points in special position . 120 3.5.1 The graph of all pseudo-triangulations of A .......... 120 3.5.2 The case with only interior collinearities . 123 3.5.3 Boundary collinearities . 128 4 Planar minimally rigid graphs and pseudo-triangulations 131 4.1 Introduction . 131 4.2 Embedding combinatorial pseudo-triangulations . 132 4.3 3-connectedness of G ........................... 137 4.4 Laman graphs admit c.p.p.t. labelings. 140 4.5Infinitesimally rigid graphs admit c.p.t. labelings. 145 5 Rigidity circuits, reciprocal diagrams and pseudo-triangulations 149 5.1 Introduction . 149 5.2 Non-crossing spherical frameworks with non-crossing reciprocals . 152 5.3 Proof of Theorems 5.2.1 and 5.2.3 . 154 5.4 Rigidity circuits. The case of pseudo-triangulations . 156 Agradecimientos Quiero expresar aqu´ı mi gratitud a todas las personas que de una u otra forma me han apoyado durante la realizaci´on de este trabajo. Muy especialmente a mis padres, pues entre lo mucho que les debo est´a el haber puesto siempre la mejor de sus voluntades para allanar el camino de la vida acad´emica que ahora termino. Tanto ellos como mi hermano aportaron palabras de aliento al d´ıaad´ıa de una investigaci´on que muchas veces les era lejana. Adem´as de su apoyo, a Encarna le agradezco tambi´en de una forma especial su compa˜n´ıa durante este tiempo. Mi reconocimiento es tambi´en para aquellos amigos y compa˜neros que, cada uno en su medida, colaboraron y pusieron en tantas ocasiones un necesario par´entesis en mi trabajo. Asimismo, quiero hacer constar mi gratitud por el apoyo, a menudo incomparable en lo profesional y personal, de Paco Santos, sin cuya gu´ıa este tra- bajo habr´ıa resultado a buen seguro mucho menos llevadero. Otros miembros del departamento, desde la entrada en esta facultad hasta hoy, facilitaron tambi´en mi andadura y me aportaron en ocasiones sus opiniones y consejos, por lo que les estoy agradecido. Finalmente, quiero mencionar a: G¨unter Ziegler, en cuyo grupo de la Technische Universit¨at Berlin realic´e una estancia en oto˜no de 2000. Emo Welzl y J¨urgen Richter-Gebert, que permitieron mi asistencia a sus cursos en el ETH de Z¨urich. Komei Fukuda y Marc Noy, quienes accedieron a recomendar mi investigaci´on. Jes´us de Loera, por su calurosa acogida durante mi estancia en la University of California at Davis en el oto˜no de 2001, y por ayudarme a intentar mejorar algunos c´alculos de esta tesis e invitarme a impartir una conferencia en Agosto de 2002. Ferr´an Hurtado, a cuya iniciativa debo tanto mi inclusi´on en una acci´on integrada de la Universitat Polit`ecnica de Catalunya como la solicitud de otras ayudas. Franz Aurenhammer, por aceptar mi estancia en la Technische Universit¨at Graz en el pr´oximo oto˜no. G¨unter Rote, que colabor´o en parte de los resultados de esta tesis. Ruth Haas, Brigitte Servatius, Hermann Servatius, Diane Souvaine, Ileana Streinu y Walter Whiteley tambi´en aportaron conversacionesutiles ´ a algunos de estos resultados. iii iv Pre´ambulo Esta tesis est´a dividida en dos partes independientes, aunque ambas proporcionan construcciones en combinatoria geom´etrica. En la primera parte, se aborda el estudio de m´etodos para construir triangulaciones “sencillas” de hipercubos de dimensi´on alta. Se ha demostrado que la obtenci´on de triangulaciones eficientes del d-cubo regular Id =[0, 1]d tiene aplicaciones de inter´es como el c´alculo de puntos fijos o la resoluci´on de ecuaciones diferenciales por m´etodos de elementos finitos. En particular, se ha dedicado especial atenci´on a tratar de determinar la triangulaci´on m´as peque˜na (en n´umero de s´ımplices 1 ) o, al menos, el comportamiento asint´otico de ese n´umero m´ınimo de s´ımplices. Los resultados previos se deben a Sallee (1984) y Haiman (1991). En el Cap´ıtulo 2 desarrollamos una construcci´on para obtener triangulaciones eficientes del producto de dos politopos generales. Aplicado al caso particular del producto de dos cubos, demostramos que este m´etodo mejora las cotas obtenidas por Haiman para el tama˜no asint´otico de una triangulaci´on del d-cubo. Nuestra construcci´on comienza con una triangulaci´on “eficiente” del producto entre un cubo y un s´ımplice de dimensiones peque˜nas. Este trabajo tiene una parte puramente te´orica (c´omo usar las triangulaciones de dimensi´on baja en objetos de dimensi´on alta) y una parte computacional (el c´alculo de la triangulaci´onoptima, ´ para nuestros prop´ositos, de los objetos de dimensi´on baja). La segunda parte de la tesis trata sobre las relaciones entre grafos planos, rigidez y pseudo-triangulaciones 2 de un conjunto de puntos en el plano. Estas´ tienen ricas propiedades combinatorias y han sido aplicadas con un amplia variedad de prop´ositos (planificaci´on de movimientos, visibilidad, estructuras de datos cin´eticas) durante la ultima´ d´ecada. En el Cap´ıtulo 3 consideramos un conjunto A de puntos en el plano y, utilizando propiedades que relacionan las pseudo-triangulaciones con la teor´ıa cl´asica de la rigidez (algunos de cuyos precursores son Euler y Cauchy), construimos un politopo simple cuyos v´ertices son todas las posibles pseudo-triangulaciones de A. Al contrario que las construcciones similares propuestas previamente para triangu- laciones, podemos describir completamente las ecuaciones de las facetas de nuestro politopo, lo que implica que se puede utilizar programaci´on lineal para optimizar funcionales sobre ´el. Por otro lado, la estructura de caras del politopo es esencial- mente la estructura de todos los grafos sin cruces que es posible dibujar en A.La 1Un d-s´ımplice es la envolvente convexa de d + 1 puntos af´ınmente independientes. 2Un pseudo-tri´angulo es un pol´ıgono simple con exactamente tres v´ertices convexos. v vi construcci´on del politopo sirve incluso para un conjunto de puntos en posici´on no general, lo que conduce a una definici´on natural de pseudo-triangulaci´on para ese caso. En cuanto a sus relaciones con rigidez, Streinu (2000) demostr´o que las pseudo- triangulaciones puntiagudas 3 son grafos planos minimalmente r´ıgidos. Demostramos que el inverso es tambi´en cierto: todo grafo plano minimalmente r´ıgido admite una inmersi´on como pseudo-triangulaci´on puntiaguda en R2, incluso bajo ciertas res- tricciones combinatorias como la imposici´on a priori de los tresangulos ´ convexos de cada cara. Este tipo de problemas de inmersi´on de grafos han sido ampliamente estudiados en la literatura, y nuestro resultado proporciona una caracterizaci´on de los grafos planos minimalmente r´ıgidos que se a˜nade a la debida a Laman.
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