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Geometric Representations of Graphs

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Geometric Representations of Graphs 1 Geometric Representations of Graphs Laszl¶ o¶ Lovasz¶ Institute of Mathematics EÄotvÄosLor¶andUniversity, Budapest e-mail: [email protected] December 11, 2009 2 Contents 0 Introduction 9 1 Planar graphs and polytopes 11 1.1 Planar graphs .................................... 11 1.2 Planar separation .................................. 15 1.3 Straight line representation and 3-polytopes ................... 15 1.4 Crossing number .................................. 17 2 Graphs from point sets 19 2.1 Unit distance graphs ................................ 19 2.1.1 The number of edges ............................ 19 2.1.2 Chromatic number and independence number . 20 2.1.3 Unit distance representation ........................ 21 2.2 Bisector graphs ................................... 24 2.2.1 The number of edges ............................ 24 2.2.2 k-sets and j-edges ............................. 27 2.3 Rectilinear crossing number ............................ 28 2.4 Orthogonality graphs ................................ 31 3 Harmonic functions on graphs 33 3.1 De¯nition and uniqueness ............................. 33 3.2 Constructing harmonic functions ......................... 35 3.2.1 Linear algebra ............................... 35 3.2.2 Random walks ............................... 36 3.2.3 Electrical networks ............................. 36 3.2.4 Rubber bands ................................ 37 3.2.5 Connections ................................. 37 4 Rubber bands 41 4.1 Rubber band representation ............................ 41 3 4 CONTENTS 4.2 Rubber bands, planarity and polytopes ...................... 43 4.2.1 How to draw a graph? ........................... 43 4.2.2 How to lift a graph? ............................ 46 4.2.3 How to ¯nd the starting face? ....................... 51 4.3 Rubber bands and connectivity .......................... 52 4.3.1 Convex embeddings ............................ 52 4.3.2 Degeneracy: essential and non-essential . 53 4.3.3 Connectivity and degeneracy ....................... 54 4.3.4 Turning the method to an algorithm ................... 57 4.4 Repulsive springs and approximating maximum cut . 60 5 Rigidity 63 5.1 Stresses ....................................... 63 5.2 Rigidity ....................................... 64 5.2.1 In¯nitesimal motions ............................ 64 5.2.2 Rigidity of convex 3-polytopes ...................... 67 5.2.3 Generic rigidity ............................... 67 6 Representing graphs by touching domains 73 6.1 Coin representation ................................. 73 6.1.1 Koebe's theorem .............................. 73 6.1.2 Formulation in the space .......................... 74 6.1.3 Preparation for the proof ......................... 74 6.1.4 The range of defects ............................ 77 6.1.5 An algorithmic proof ............................ 81 6.1.6 *Another algorithmic proof ........................ 82 6.1.7 *From rubber bands to touching circles . 84 6.1.8 Conformal transformations ........................ 85 6.1.9 Applications of circle packing ....................... 86 6.1.10 Circle packing and the Riemann Mapping Theorem . 89 6.2 *Extensions ..................................... 90 6.2.1 Orthogonal circles ............................. 90 6.2.2 Tangency graphs of general convex domains . 92 6.3 Square tilings .................................... 93 6.3.1 Current flow through a rectangle ..................... 93 6.3.2 Tangency graphs of square tilings ..................... 95 CONTENTS 5 7 Analytic functions on graphs 101 7.1 Circulations and homology ............................ 101 7.2 Discrete holomorphic forms from harmonic functions . 102 7.3 Operations ..................................... 104 7.3.1 Integration ................................. 104 7.3.2 Critical analytic functions . 105 7.3.3 Polynomials, exponentials and approximation . 106 7.4 Nondegeneracy properties of rotation-free circulations . 107 7.4.1 Rubber bands and analytic functions . 111 7.5 Geometric representations and discrete analytic functions . 111 7.5.1 Square tilings and analytic functions . 111 7.5.2 Circle packings and analytic functions . 113 7.5.3 Touching polygon representations . 113 7.6 Novikov's discrete analytic functions . 114 7.7 Discrete analytic functions from circle packings . 114 8 The Colin de Verdi`ereNumber 115 8.1 The de¯nition .................................... 115 8.1.1 Motivation ................................. 115 8.1.2 Formal de¯nition .............................. 116 8.1.3 The Strong Arnold Property . 116 8.2 Basic properties ................................... 118 8.3 Small values ..................................... 118 8.4 Nullspace representation .............................. 119 8.4.1 A nodal theorem for Colin de Verdi`erematrices . 119 8.4.2 Steinitz representations and Colin de Verdi`erematrices . 121 8.5 Gram representation ................................ 121 8.6 Related graph parameters ............................. 121 9 Orthogonal representations 123 9.1 Orthogonal representations: de¯nition . 123 9.2 Smallest cone and the theta function . 124 9.2.1 Shannon capacity .............................. 125 9.2.2 De¯nition and basic properties of the theta function . 126 9.2.3 More expressions for # . 129 9.2.4 More properties ............................... 132 9.2.5 How good is # as an approximation? . 135 9.2.6 Perfect graphs ............................... 137 9.2.7 The TSTAB body and weighted θ-function . 138 6 CONTENTS 9.3 Minimum dimension ................................ 139 9.3.1 Minimum dimension with no restrictions . 139 9.3.2 General position orthogonal representations . 140 9.3.3 Faithful orthogonal representations . 143 9.3.4 Orthogonal representations with the Strong Arnold Property . 144 9.4 The variety of orthogonal representations . 147 9.5 Related representations .............................. 152 10 Graph independence to linear independence 153 10.1 Cover-critical and independence-critical graphs . 153 11 Metric embeddings 157 11.1 Embeddings in low dimension . 157 11.2 Embeddings with small distortion . 158 11.3 Application to multicommodity flows . 160 11.4 Volume-respecting embeddings . 161 12 Adjacency matrix and regularity partitions 163 12.1 Similarity metric .................................. 163 12.2 Regularity partitions ................................ 163 13 Some general issues 165 13.1 Non-degeneracy ................................... 165 13.2 Duality ....................................... 166 13.3 Algorithms ..................................... 166 13.4 Background from linear algebra . 169 13.4.1 Basic facts about eigenvalues . 169 13.4.2 Semide¯nite matrices . 170 13.4.3 Cross product ................................ 172 13.5 Graph theory .................................... 173 13.5.1 Basics .................................... 173 13.5.2 Szemer¶edipartitions ............................ 173 13.6 Eigenvalues of graphs ............................... 173 13.6.1 Matrices associated with graphs . 173 13.6.2 The largest eigenvalue . 175 13.6.3 The smallest eigenvalue . 176 13.6.4 The eigenvalue gap ............................. 178 13.6.5 The number of di®erent eigenvalues . 186 13.6.6 Spectra of graphs and optimization . 188 13.7 Convex polytopes .................................. 189 CONTENTS 7 13.7.1 Polytopes and polyhedra . 189 13.7.2 The skeleton of a polytope . 190 13.7.3 Polar, blocker and antiblocker . 191 13.7.4 Optimization ................................ 192 13.8 Semide¯nite optimization ............................. 192 13.8.1 Semide¯nite programs . 193 13.8.2 Fundamental properties of semide¯nite programs . 194 13.8.3 Algorithms for semide¯nite programs . 196 Bibliography 199 8 CONTENTS Chapter 0 Introduction 9 10 CHAPTER 0. INTRODUCTION Chapter 1 Planar graphs and polytopes 1.1 Planar graphs A graph G = (V; E) is planar, if it can be drawn in the plane so that its edges are Jordan curves and they intersect only at their endnodes1.A plane map is a planar graph with a ¯xed embedding. We also use this phrase to denote the image of this embedding, i.e., the subset of the plane which is the union of the set of points representing the nodes and the Jordan curves representing the edges. The complement of a plane map decomposes into a ¯nite number of arcwise connected pieces, which we call the faces (or countries) of the planar map. We usually denote the number of nodes, edges and faces of a planar graph by n, m and f. Every planar map G = (V; E) has a dual map G¤ = (V ¤;E¤) (Figure 1.1) As an abstract graph, this can be de¯ned as the graph whose nodes are the faces of G. If the two faces share k edges, then we connect them in G¤ by k edges, so that each edge e 2 E will correspond to an edge e¤ of G¤. So jE¤j = jEj. Figure 1.1: A planar map and its dual. This dual has a natural drawing in the plane: in the interior of each face F of G we select 1We use the word node for the node of a graph, the word vertex for the vertex of a polytope, and the word point for points in the plane or in other spaces. 11 12 CHAPTER 1. PLANAR GRAPHS AND POLYTOPES a point vF (which can be called its capital if we use the country terminology), and on each ¤ edge e 2 E we select a point ue (this will not be a node of G , just an auxiliary point). We connect vF to the points ue for each edge on the boundary of F by nonintersecting Jordan curves inside F . If the boundary of F goes through e twice (i.e., both sides of e belong to F ), then we connect vF to ue by two curves, entering e from two sides. The
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