Geometric Representations of Graphs with low Polygonal Complexity vorgelegt von Diplom-Mathematiker Torsten Ueckerdt aus Berlin Von der Fakultät II – Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften – Dr. rer. nat. – genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. Jörg Liesen Berichter: Prof. Dr. Stefan Felsner Prof. Dr. Jan Kratochvil Prof. Dr. Stephen G. Kobourov Tag der wissenschaftlichen Aussprache: 4. November 2011 Berlin 2012 D 83 Preface This thesis is the fruit of my time in the Discrete Mathematics Group at TU Berlin. This group is a particularly great culture medium for mathematical plants of all kinds. Someone throws in a seed in form of a problem, a thought or an idea, and everybody can watch it either flourishing or wilting. Right from the beginning I was pleased about the possibility to work and learn within the Discrete Mathematics Group, and today I can take some nice flowers along. I want to thank my advisor Stefan Felsner for always letting me choose my prob- lems, for joining their treatment with well-trained patience and valuable ideas, and for his support of any kind. I am also thankful to Jan Kratochvil and Stephen G. Kobourov for being reviewers of my thesis. During the years, I had the opportunity to meet and work with a lot of smart and kind researchers, in particular Marie Albenque, Daniel Heldt, Andrea Hoffkamp, and Kolja Knauer, as well as, Thomas Hixon and Irina Mustata here in Berlin, Bartłomiej Bosek, Tomasz Krawczyk, Piotr Micek, and Bartosz Walczak from Jagiellonian Uni- versity in Kraków, Poland, and Stephen G. Kobourov and Muhammad Jawaherul Alam from University of Arizona. I would like to emphasize my thanks to Daniel Heldt and Kolja Knauer for our joint research for Chapter 4, but also for listening to my thoughts and discussing even the wild ones. Furthermore, I do thank Stephen G. Kobourov and Muhammad Jawaherul Alam: During their stay in Berlin we did the research for Chapter 3 in a relaxed and productive atmosphere, which I very much enjoyed. Last and not least, I very much want to thank my wife, für alle bestehenden und bestandenen Abenteuer. Thank you all! Torsten Ueckerdt Berlin, August 2011 Contents Introduction 1 1 Preliminaries 5 1.1 VertexOrderings ................................ 5 1.1.1 Degeneracy ............................... 6 1.1.2 Tree-Width............................... 7 1.1.3 Canonical Orders . 8 1.1.4 SchnyderWoods............................ 10 1.1.5 Separation-Trees and Level-i Subgraphs . 14 1.2 Orientations with Prescribed Out-Degrees . ...... 16 1.2.1 Near-Linear Time Computation . 20 1.3 Rectangle-Representations and Transversal Structures.......... 24 2 Side Contact Representations 29 2.1 Non-Rotated Representations and Schnyder Woods . ..... 35 2.1.1 Overall Complexity and Number of Segments . 49 2.2 Representations from Nesting Assignments . ..... 51 2.3 Lower Bounds on the Complexity . 58 3 Cartograms 63 3.1 Area-Universal Layouts . 67 3.2 Cartograms for Hamiltonian Maximally Planar Graphs . ...... 75 i Contents 3.2.1 One-Sided Hamiltonian Cycles . 81 3.3 Lower Bounds on the Complexity . 86 3.3.1 Hole-Free Cartograms for Planar 3-Trees . 88 3.4 Tackling 4-Connected Maximally Planar Graphs . ..... 93 4 Edge-IntersectionGraphsofGridPaths 99 4.1 The Bend-Number of Complete Bipartite Graphs . 103 4.2 The Bend-Number of Planar and Outer-Planar Graphs . 112 4.3 Fixed Degeneracy, Tree-Width, or Maximum Degree . 126 4.3.1 The Bend-Number in Terms of the Degeneracy . 127 4.3.2 The Bend-Number in Terms of the Tree-Width . 129 4.3.3 The Bend-Number in Terms of the Maximum Degree . 131 4.4 Recognizing Single-Bend Graphs is NP-Complete . 132 4.4.1 ClauseGadgets ............................ 133 4.4.2 TheReduction............................. 135 4.5 Comparison with Interval-Number and (Local) Track-Number . 138 Open Questions 143 Bibliography 147 Index 157 ii Introduction There are several ways to think of a graph and many of them involve drawing pictures. In the most classical visualization vertices are considered as points in the plane and edges as continuous curves connecting two points, such as in the top-left of Figure 1. Indeed, graph properties of eminent importance, e.g., planarity, are defined with respect to those drawings. Other popular graph visualizations include intersection representations. For exam- ple, every vertex is depicted as a point set in the plane and an edge between two vertices is described by an intersection of the corresponding point sets, such as in the bottom-left of Figure 1. In a contact representation the point set for each vertex is compact and those sets are pairwise interior disjoint. Then intersections involve only boundaries, as in the right of Figure 1, and are thus called contacts. Figure 1: A drawing, an intersection, and a contact representation of a graph. 1 Introduction Many kinds of intersection graphs have been considered, ranging back from Koebe’s “Kissing Coins Theorem” [Koe36] in 1936, up to segment representations of planar graphs due to Chapolin and Gonçalves in 2009 [CG09], and further. Within this thesis we investigate two types of intersection graphs, in both of which vertices are represented by polygonal objects in the plane. We measure the com- plexity of a polygonal object by the number of its corners. We are then particularly interested in a low polygonal complexity for every vertex, i.e., we want the maximum complexity over all vertices to be as low as possible. Chapter 2 deals with side contact representations with one simple polygon for every vertex. Every graph that admits such a representation is necessarily planar. The major part of Chapter 2 concerns hole-free rectilinear representations, i.e., those in which every side of every polygon is either horizontal or vertical, and where the union of all polygons does not leave any holes. The right of Figure 2 shows such a contact representation. We consider here maximally planar graphs only, which is a natural (and almost necessary) assumption in this setting. One of our results is a new proof that polygons of complexity 8 are always sufficient and sometimes necessary for a hole-free rectilinear representation of a maximally planar graph. In Chapter 3 we investigate what happens if we additionally prescribe the area of each and every polygon in the representation. A representation that respects a set of desired areas is known as a cartogram. For example, we prove that one can require any set of areas without increasing the worst-case maximum complexity of a hole-free rectilinear representation, i.e., 8-gons are still sufficient for every cartogram of a maximally planar graph. The second type of intersection graph is investigated in Chapter 4. In an EPG representation vertices are represented as polygonal paths with solely horizontal and vertical segments, and an edge occurs whenever two paths overlap along some part of non-zero length, i.e., neither a touching point nor a crossing causes an edge. An example of such a representation is provided in the left of Figure 2. This time, every graph admits an EPG representation. However, we again want the polygonal com- plexity, i.e., the number of corners, per path to be low. The least possible maximum complexity over all paths for a given graph is the bend-number. We give several new upper and lower bounds on the maximum bend-number for certain graph classes, such as, planar and outer-planar graphs, complete bipartite graphs, graphs of certain tree-width, maximum degree, or degeneracy. This is how the thesis is organized. Chapter 1: This chapter introduces the basic concepts and notation we use within this thesis. Section 1.1 is about vertex orderings and in particular building 2 Figure 2: A planar graph with an EPG representation on the left and a side contact representation on the right. sequences associated with them. Those building sequences underlie many of the constructive proofs presented in subsequent chapters. We define the degeneracy and the tree-width of a graph in terms of vertex orderings. For maximally planar graphs, we review the concepts of canonical orders and Schnyder woods, and outline some aspects of their close relation to each other. Furthermore, we define the separation-tree of an embedded maximally planar graph and deduce the level-i subgraphs from it. In Section 1.2 we consider orientations with prescribed out-degrees, so-called α-orientations, review their most important properties and present an algorithm that computes the minimal α-orientation in near-linear time. We close the preliminaries with Section 1.3, in which we briefly introduce rectangle-representations and transversal structures. Chapter 2: In this chapter we investigate side contact representations of planar graphs, i.e., vertices are represented by simple polygons which are pairwise in- terior disjoint, and edges correspond to side contacts. In Section 2.1 we are particularly interested in rectilinear hole-free representations with low polygo- nal complexity. We present a general method to obtain such a representation for a maximally planar graph from a Schnyder wood. In special cases, we obtain a characterization of those maximally planar graphs that admit a non-rotated rectilinear representation with complexity 4, and 6. Furthermore, we derive a new compact floor plan for maximally planar graphs. In Section 2.2 we improve a result of Sun and Sarrafzadeh [SS93] by presenting a linear-time algorithm that constructs a rectilinear representation with complexity 6 based on a nesting assignment. Our algorithm can be adjusted to construct non-rectilinear repre- sentations of complexity 5, which in particular proves their existence under the presence of a nesting assignment. At the end of this chapter in Section 2.3, we provide a general method to compute lower bounds on the complexity of side contact representations.