Colouring Contact Graphs of Squares and Rectilinear Polygons
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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. -
Representing Graphs by Polygons with Edge Contacts in 3D
Representing Graphs by Polygons with Side Contacts in 3D∗ Elena Arseneva1, Linda Kleist2, Boris Klemz3, Maarten Löffler4, André Schulz5, Birgit Vogtenhuber6, and Alexander Wolff7 1 Saint Petersburg State University, Russia [email protected] 2 Technische Universität Braunschweig, Germany [email protected] 3 Freie Universität Berlin, Germany [email protected] 4 Utrecht University, the Netherlands [email protected] 5 FernUniversität in Hagen, Germany [email protected] 6 Technische Universität Graz, Austria [email protected] 7 Universität Würzburg, Germany orcid.org/0000-0001-5872-718X Abstract A graph has a side-contact representation with polygons if its vertices can be represented by interior-disjoint polygons such that two polygons share a common side if and only if the cor- responding vertices are adjacent. In this work we study representations in 3D. We show that every graph has a side-contact representation with polygons in 3D, while this is not the case if we additionally require that the polygons are convex: we show that every supergraph of K5 and every nonplanar 3-tree does not admit a representation with convex polygons. On the other hand, K4,4 admits such a representation, and so does every graph obtained from a complete graph by subdividing each edge once. Finally, we construct an unbounded family of graphs with average vertex degree 12 − o(1) that admit side-contact representations with convex polygons in 3D. Hence, such graphs can be considerably denser than planar graphs. 1 Introduction A graph has a contact representation if its vertices can be represented by interior-disjoint geometric objects1 such that two objects touch exactly if the corresponding vertices are adjacent. -
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]. -
Square-Contact Representations of Partial 2-Trees and Triconnected Simply-Nested Graphs∗†
Square-Contact Representations of Partial 2-Trees and Triconnected Simply-Nested Graphs∗† Giordano Da Lozzo1, William E. Devanny2, David Eppstein3, and Timothy Johnson4 1 Computer Science Department, University of California, Irvine, USA [email protected] 2 Computer Science Department, University of California, Irvine, USA [email protected] 3 Computer Science Department, University of California, Irvine, USA [email protected] 4 Computer Science Department, University of California, Irvine, USA [email protected] Abstract A square-contact representation of a planar graph G = (V, E) maps vertices in V to interior- disjoint axis-aligned squares in the plane and edges in E to adjacencies between the sides of the corresponding squares. In this paper, we study proper square-contact representations of planar graphs, in which any two squares are either disjoint or share infinitely many points. We characterize the partial 2-trees and the triconnected cycle-trees allowing for such represen- tations. For partial 2-trees our characterization uses a simple forbidden subgraph whose structure forces a separating triangle in any embedding. For the triconnected cycle-trees, a subclass of the triconnected simply-nested graphs, we use a new structural decomposition for the graphs in this family, which may be of independent interest. Finally, we study square-contact representations of general triconnected simply-nested graphs with respect to their outerplanarity index. 1998 ACM Subject Classification G.2.2 Graph Theory Keywords and phrases Square-Contact Representations, Partial 2-Trees, Simply-Nested Graphs Digital Object Identifier 10.4230/LIPIcs.ISAAC.2017.24 1 Introduction Contact representations of graphs, in which the vertices of a graph are represented by non- overlapping or non-crossing geometric objects of a specific type, and edges are represented by tangencies or other contacts between these objects, form an important line of research in graph drawing and geometric graph theory. -
Edge-Weighted Contact Representations of Planar Graphs
Edge-weighted contact representations of planar graphs Martin Nollenburg,¨ Roman Prutkin, and Ignaz Rutter Institut fur¨ Theoretische Informatik, Karlsruhe Institute of Technology, Germany Abstract. We study contact representations of edge-weighted planar graphs, where vertices are rectangles or rectilinear polygons and edges are polygon con- tacts whose lengths represent the edge weights. We show that for any given edge-weighted planar graph whose outer face is a quadrangle, that is internally triangulated and that has no separating triangles we can construct in linear time an edge-proportional rectangular dual if one exists and report failure otherwise. For a given combinatorial structure of the contact representation and edge weights interpreted as lower bounds on the contact lengths, a corresponding contact repre- sentation that minimizes the size of the enclosing rectangle can be found in linear time. If the combinatorial structure is not fixed, we prove NP-hardness of deciding whether a contact representation with bounded contact lengths exists. Finally, we give a complete characterization of the rectilinear polygon complexity required for representing biconnected internally triangulated graphs: For outer- planar graphs complexity 8 is sufficient and necessary, and for graphs with two adjacent or multiple non-adjacent internal vertices the complexity is unbounded. 1 Introduction Representing graphs by intersections or contacts of geometric objects has a long history in graph theory and graph drawing, which is covered in monographs and surveys [12,20]. For example, Koebe’s circle packing theorem from 1936 establishes that every planar graph has a contact representation by touching disks (and vice versa) [15]; more recently it was shown that every planar graph is the intersection graph of line segments [6]. -
Square-Contact Representations of Partial 2-Trees and Triconnected Simply-Nested Graphs∗
Square-Contact Representations of Partial 2-Trees and Triconnected Simply-Nested Graphs∗ Giordano Da Lozzo, William E. Devanny, David Eppstein, and Timothy Johnson Computer Science Department, University of California, Irvine, USA {gdalozzo,wdevanny,eppstein,tujohnso}@uci.edu Abstract A square-contact representation of a planar graph G = (V, E) maps vertices in V to interior- disjoint axis-aligned squares in the plane and edges in E to adjacencies between the sides of the corresponding squares. In this paper, we study proper square-contact representations of planar graphs, in which any two squares are either disjoint or share infinitely many points. We characterize the partial 2-trees and the triconnected cycle-trees allowing for such represent- ations. For partial 2-trees our characterization uses a simple forbidden subgraph whose structure forces a separating triangle in any embedding. For the triconnected cycle-trees, a subclass of the triconnected simply-nested graphs, we use a new structural decomposition for the graphs in this family, which may be of independent interest. Finally, we study square-contact representations of general triconnected simply-nested graphs with respect to their outerplanarity index. 1998 ACM Subject Classification G.2.2 Graph Theory Keywords and phrases Square-Contact Representations, Partial 2-Trees, Simply-Nested Graphs 1 Introduction Contact representations of graphs, in which the vertices of a graph are represented by non- overlapping or non-crossing geometric objects of a specific type, and edges are represented by tangencies or other contacts between these objects, form an important line of research in graph drawing and geometric graph theory. For instance, the Koebe–Andreev–Thurston circle packing theorem states that every planar graph is a contact graph of circles [12]. -
Geometric Separator Theorems and Applications - DRAFT
Smith & Wormald typeset 849 Jul 2, 2020 Separator theorems Geometric separator theorems and applications - DRAFT Warren D. Smith∗ Nicholas C. Wormaldy [email protected], [email protected] Abstract | 2.2 k-Outerplanar graphs . 6 We find a large number of \geometric separator 2.3 Chordal graphs . 6 theorems" such as: I: Given N disjoint iso-oriented 2.4 Series Parallel graphs . 6 squares in the plane, there exists a rectangle with 2.5 Algebraic graphs . 6 ≤ 2N=3 squaresp inside, ≤ 2N=3 squares outside, and ≤ (4 + o(1)) N partly in & out. II: There exists a 2.6 Connection to eigenvalues; cutting mani- folds . 6 rectangle that is crossed by the minimal spanningp tree of N sites in the plane at ≤ (4 · 31=4 + o(1)) N 2.7 Planar graphs . 7 points, having ≤ 2N=3 sites inside and outside. These 2.8 Computational complexity of finding op- theorems yield a large number of applications, such timal and near optimal cuts and edge sep- as subexponential algorithms for traveling salesman arators . 8 tour and rectilinear Steiner minimal tree, new point 2.9 Graphs of low genus, or with excluded mi- location algorithms, and new upper and lower bound nors . 8 proofs for \planar separator theorems." We also sur- 2.10 Other . 9 vey graph separator theorems and study the problem of covering convex bodies with smaller copies. 3 Covering convex bodies with smaller Keywords | Separator theorems, Steiner minimal tree, Rec- copies 9 tilinear Steiner minimal tree, traveling salesman tour, minimum 3.1 Smaller scaled translated copies . -
Equiangular Polygon Contact Representations⋆
Equiangular polygon contact representations? Stefan Felsner??;1, Hendrik Schrezenmaier??;1(B), and Raphael Steiner2 1 Institut f¨urMathematik, Technische Universit¨atBerlin, Berlin, Germany ffelsner,[email protected] 2 Fachgebiet Mathematik, FernUniversit¨at in Hagen, Hagen, Germany [email protected] Abstract. Planar graphs are known to have contact representations of various types. The most prominent example is Koebe's `kissing coins theorem'. Its rediscovery by Thurston lead to effective versions of the Riemann Mapping Theorem and motivated Schramm's Monster Packing Theorem. Monster Packing implies the existence of contact representa- tions of planar triangulations where each vertex v is represented by a homothetic copy of some smooth strictly-convex prototype Pv. With this work we aim at computable approximations of Schramm rep- resentations. For fixed K approximate Pv by an equiangular K-gon Qv with horizontal basis. From Schramm's work it follows that the given triangulation also has a contact representation with homothetic copies of these K-gons. Our approach starts by guessing a K-contact-structure, i.e., the combinatorial structure of a contact representation. From the combinatorial data, we build a system of linear equations whose variables correspond to lengths of boundary segments of the K-gons. If the system has a non-negative solution, this yields the intended contact representa- tion. If the solution of the system contains negative variables, these can be used as sign-posts indicating how to change the K-contact-structure for another try. In the case K = 3 the K-contact-structures are Schnyder woods, in the case K = 4 they are transversal structures. -
Vertex Contact Graphs of Paths on a Grid
Vertex Contact Graphs of Paths on a Grid Nieke Aerts and Stefan Felsner∗ faerts,[email protected] Technische Universit¨atBerlin Institut f¨urMathematik Strasse des 17. Juni 136 10623 Berlin, Germany Abstract Contact and intersection representations of graphs and particularly of planar graphs have been studied for decades. The by now best known result in the area may be the Koebe-Andreev-Thurston circle packing theorem. A more recent highlight in the area is a result of Chalopin and Gon¸calves: every planar graph is an intersection graph of segments in the plane. This boosted the study of intersection and contact graphs of restricted classes of curves. In this paper we study planar graphs that are VCPG, i.e. graphs admitting a representation as Vertex Contact graph of Paths on a Grid. In such a representation the vertices of G are represented by a family of interiorly disjoint grid-paths. Adjacencies are represented by contacts between an endpoint of one grid-path and an interior point of another grid-path. Defining u ! v if the path of u ends on path of v we obtain an orientation on G from a VCPG representation. To get hand on the bends of the grid path the 2-orientation is not enough. We therefore consider pairs (α; ): a 2-orientation α and a flow in the angle graph. The 2-orientation describes the contacts of the ends of a grid-path and the flow describes the behavior of a grid-path between its two ends. We give a necessary and sufficient condition for such a pair (α; ) to be realizable as a VCPG. -
Representing Graphs and Hypergraphs by Touching Polygons in 3D William Evans 1 Paweł Rzążewski 2,3 Noushin Saeedi 1 Chan-Su Shin 4 Alexander Wolff 5
Representing Graphs and Hypergraphs by Touching Polygons in 3D William Evans 1 Paweł Rzążewski 2;3 Noushin Saeedi 1 Chan-Su Shin 4 Alexander Wolff 5 1University of British Columbia, Vancouver, Canada 2Warsaw University of Technology, Faculty of Mathematics and Information Science, Warszawa, Poland 3Institute of Informatics, University of Warsaw, Warszawa, Poland 4Hankuk University of Foreign Studies, Yongin, Republic of Korea 5Universität Würzburg, Würzburg, Germany Dedicated to Honza Kratochvíl on his 60th birthday. Abstract Contact representations of graphs have a long history. Most research has focused on problems in 2D, but 3D contact representations have also been investigated, mostly concerning fully-dimensional geometric objects such as spheres or cubes. In this paper we study contact representations with convex polygons in 3D. We show that every graph admits such a rep- resentation. Since our representations use super-polynomial coordinates, we also construct representations on grids of polynomial size for specific graph classes (bipartite, subcubic). For hypergraphs, we represent their duals, that is, each vertex is represented by a point and each edge by a polygon. We show that even regular and quite small hypergraphs do not admit such representations. On the other hand, the two smallest Steiner triple systems can be represented. arXiv:1908.08273v3 [cs.CG] 31 Mar 2020 A preliminary version of this work appeared in Proc. 27th Int. Symp. Graph Drawing & Network Vis. (GD 2019) [13]. W.E. and N.S. were funded by an NSERC Discovery grant and in part by the Institute for Computing, Information and Cognitive Systems (ICICS) at UBC. P.Rz. was supported by the ERC starting grant CUTACOMBS (no. -
Classifying Convex Bodies by Their Contact and Intersection Graphs
Classifying Convex Bodies by Their Contact and Intersection Graphs Anders Aamand ! BARC, University of Copenhagen, Denmark Mikkel Abrahamsen ! BARC, University of Copenhagen, Denmark Jakob Bæk Tejs Knudsen ! BARC, University of Copenhagen, Denmark Peter Michael Reichstein Rasmussen ! BARC, University of Copenhagen, Denmark Abstract Let A be a convex body in the plane and A1,...,An be translates of A. Such translates give rise to an intersection graph of A, G = (V, E), with vertices V = {1, . , n} and edges E = {uv | Au ∩ Av ̸= ∅}. The subgraph G′ = (V, E′) satisfying that E′ ⊂ E is the set of edges uv for which the interiors of ′ Au and Av are disjoint is a unit distance graph of A. If furthermore G = G, i.e., if the interiors of Au and Av are disjoint whenever u ̸= v, then G is a contact graph of A. In this paper, we study which pairs of convex bodies have the same contact, unit distance, or intersection graphs. We say that two convex bodies A and B are equivalent if there exists a linear transformation B′ of B such that for any slope, the longest line segments with that slope contained in A and B′, respectively, are equally long. For a broad class of convex bodies, including all strictly convex bodies and linear transformations of regular polygons, we show that the contact graphs of A and B are the same if and only if A and B are equivalent. We prove the same statement for unit distance and intersection graphs. 2012 ACM Subject Classification Theory of computation → Computational geometry; Mathematics of computing → Graph theory; Mathematics of computing → Discrete mathematics Keywords and phrases convex body, contact graph, intersection graph Digital Object Identifier 10.4230/LIPIcs.SoCG.2021.3 Related Version Full Version: https://arxiv.org/abs/1902.01732 [1] Funding The authors are part of BARC, Basic Algorithms Research Copenhagen, supported by the VILLUM Foundation grant 16582. -
Recognition of Unit Disk Graphs for Caterpillars, Embedded Trees, and Outerplanar Graphs
Unit Disk Representations of Embedded Trees, Outerplanar and Multi-Legged Graphs Sujoy Bhore1[0000−0003−0104−1659], Maarten L¨offler2, Soeren Nickel3[0000−0001−5161−3841], and Martin N¨ollenburg3[0000−0003−0454−3937] 1 Indian Institute of Science Education and Research, Bhopal, India [email protected] 2 Department of Computing and Information Sciences, Utrecht University [email protected] 3 Algorithms and Complexity Group, TU Wien, Vienna, Austria [soeren.nickel|noellenburg]@ac.tuwien.ac.at Abstract. A unit disk intersection representation (UDR) of a graph G represents each vertex of G as a unit disk in the plane, such that two disks intersect if and only if their vertices are adjacent in G.A UDR with interior-disjoint disks is called a unit disk contact representation (UDC). We prove that it is NP-hard to decide if an outerplanar graph or an embedded tree admits a UDR. We further provide a linear-time decidable characterization of caterpillar graphs that admit a UDR. Finally we show that it can be decided in linear time if a lobster graph admits a weak UDC, which permits intersections between disks of non-adjacent vertices. 1 Introduction The representation of graphs as contacts or intersections of disks has been a major topic of investigation in geometric graph theory and graph drawing. The famous circle packing theorem states that every planar graph has a contact representation by touching disks (of various size) and vice versa [13]. Since then, a large body of research has been devoted to the representation of planar graphs as contacts or intersections of various kinds of geometric objects [4,5,8,9].