DOCSLIB.ORG

The Number of Edges in $K$-Quasi-Planar Graphs.” SIAM Journal on Discrete Mathematics 27, No

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Number of Edges in $K$-Quasi-Planar Graphs.” SIAM Journal on Discrete Mathematics 27, No The Number of Edges in k-Quasi-planar Graphs The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Fox, Jacob, János Pach, and Andrew Suk. “The Number of Edges in $k$-Quasi-planar Graphs.” SIAM Journal on Discrete Mathematics 27, no. 1 (March 19, 2013): 550-561. © 2013, Society for Industrial and Applied Mathematics As Published http://dx.doi.org/10.1137/110858586 Publisher Society for Industrial and Applied Mathematics Version Final published version Citable link http://hdl.handle.net/1721.1/79615 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. SIAM J. DISCRETE MATH. c 2013 Society for Industrial and Applied Mathematics Vol. 27, No. 1, pp. 550–561 THE NUMBER OF EDGES IN k-QUASI-PLANAR GRAPHS∗ † ´ ‡ † JACOB FOX ,JANOS PACH , AND ANDREW SUK Abstract. A graph drawn in the plane is called k-quasi-planar if it does not contain k pair- wise crossing edges. It has been conjectured for a long time that for every fixed k,themaximum number of edges of a k-quasi-planar graph with n vertices is O(n). The best known upper bound c n n O(log k) n n α(n) k is (log ) . In the present paper, we improve this bound to ( log )2 in the special α n case where the graph is drawn in such a way that every pair of edges meet at most once. Here ( ) denotes the (extremely slowly growing) inverse of the Ackermann function. We also make further progress on the conjecture for k-quasi-planar graphs in which every edge is drawn as an x-monotone curve. Extending some ideas of Valtr, we prove that the maximum number of edges of such graphs ck6 is at most 2 n log n. Key words. topological graphs, quasi-planar graphs, Turan-type problems AMS subject classifications. 05C35, 05C10, 68R10, 52C10 DOI. 10.1137/110858586 1. Introduction. A topological graph is a graph drawn in the plane such that its vertices are represented by points and its edges are represented by nonself-intersecting arcs connecting the corresponding points. In notation and terminology, we make no distinction between the vertices and edges of a graph and the points and arcs repre- senting them, respectively. No edge is allowed to pass through any point representing a vertex other than its endpoints. Any two edges can intersect only in a finite number of points. Tangencies between the edges are not allowed. That is, if two edges share an interior point, then they must properly cross at this point. A topological graph is simple if every pair of its edges intersect at most once: at a common endpoint or at a proper crossing. If the edges of a graph are drawn as straight-line segments, then the graph is called geometric. Finding the maximum number of edges in a topological graph with a forbidden crossing pattern is a fundamental problem in extremal topological graph theory (see [2,3,4,6,10,12,16,21,23]).ItfollowsfromEuler’spolyhedralformulathatevery topological graph on n vertices and with no two crossing edges has at most 3n − 6 edges. A graph is called k-quasi-planar if it can be drawn as a topological graph with no k pairwise crossing edges. A graph is 2-quasi-planar if and only if it is planar. According to an old conjecture (see Problem 1 in section 9.6 of [5]), for any fixed k ≥ 2 there exists a constant ck such that every k-quasi-planar graph on n vertices has at most ckn edges. Agarwal et al. [4] were the first to prove this conjecture for ∗Received by the editors December 12, 2011; accepted for publication (in revised form) October 30, 2012; published electronically March 19, 2013. A preliminary version of this paper, with A. Suk as its sole author, appears in Proceedings of the 19th International Symposium on Graph Drawing (GD 2011, TU Eindhoven), Lecture Notes in Comput. Sci. 7034, Springer-Verlag, New York, 2011, pp. 266–277. http://www.siam.org/journals/sidma/27-1/85858.html †Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02114 ([email protected], [email protected]). The first author was supported by a Simons Fellowship and NSF grant DMS 1069197. The third author was supported by an NSF Postdoctoral Fellowship. ‡EPFL Lausanne, CH-1015 Lausanne, Switzerland and Courant Institute, New York University, Downloaded 07/17/13 to 18.51.1.228. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php New York, NY 10012 ([email protected]). This author was supported by Hungarian Science Foun- dation EuroGIGA grant OTKA NN 102029, by Swiss National Science Foundation grant 200021- 125287/1, and by NSF grant CCF-08-30272. 550 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE NUMBER OF EDGES IN k-QUASI-PLANAR GRAPHS 551 simple 3-quasi-planar graphs. Later, Pach, Radoiˇci´c, and T´oth [17] generalized the result to all 3-quasi-planar graphs. Ackerman [1] proved the conjecture for k =4. For larger values of k, first Pach, Shahrokhi, and Szegedy [18] showed that every 2k−4 simple k-quasi-planar graph on n vertices has at most ckn(log n) edges. For k ≥ 3 and for all (not necessarily simple) k-quasi-planar graphs, Pach, Radoiˇci´c, and 4k− 12 T´oth [17] established the upper bound ckn(log n) . Plugging into these proofs ≥ the above mentioned result of Ackerman [1], for k 4, we obtain the slightly better 2k−8 4k−16 bounds ckn(log n) and ckn(log n) , respectively. For large values of k,the exponent of the polylogarithmic factor in these bounds was improved by Fox and Pach [10], who showed that the maximum number of edges of a k-quasi-planar graph O(log k) on n vertices is n(log n) . For the number of edges of geometric graphs, that is, graphs drawn by straight- line edges, Valtr [22] proved the upper bound O(n log n ). He also extended this result to simple topological graphs whose edges are drawn as x-monotone curves [23]. The aim of this paper is to improve the best known bound, n(log n)O(log k),onthe number of edges of a k-quasi-planar graph in two special cases: for simple topological graphs and for not necessarily simple topological graphs whose edges are drawn as x-monotone curves. In both cases, we improve the exponent of the polylogarithmic factor from O(log k)to1+o(1). Theorem 1.1. Let G =(V,E) be a k-quasi-planar simple topological graph with α n ck n vertices. Then |E(G)|≤(n log n)2 ( ) ,whereα(n) denotes the inverse of the Ackermann function and ck is a constant that depends only on k. Recall that the Ackermann (more precisely, the Ackermann–P´ eter) function A(n) is defined as follows. Let A1(n)=2n, and let Ak(n )=Ak−1(Ak(n − 1)) for k = n 2, 3,.... In particular, we have A2(n)=2 ,andA3(n) is an exponential tower of n { ≥ ≥ two’s. Now let A(n)=An(n), and let α(n) be defined as α(n)=min k 1:A(k) n}. This function grows much slower than the inverse of any primitive recursive function. Theorem 1.2. Let G =(V,E) be a k-quasi-planar (not necessarily simple) topological graph with n vertices, whose edges are drawn as x-monotone curves. Then | |≤ ck6 E(G) 2 n log n,wherec is an absolute constant. In both proofs, we follow the approach of Valtr [23] and apply results on gener- alized Davenport–Schinzel sequences. An important new ingredient of the proof of Theorem 1.1 is a corollary of a separator theorem established in [9] and developed in [8]. Theorem 1.2 is not only more general than Valtr’s result, which applies only to simple topological graphs, but its proof gives a somewhat better upper bound: we use a result of Pettie [20], which improves the dependence on k from double exponential to single exponential. 2. Generalized Davenport–Schinzel sequences. The sequence u = a ,a , 1 2 ...,am is called l-regular if any l consecutive terms are pairwise different. For integers l, t ≥ 2, the sequence S = s1,s2,...,slt of length lt is said to be of type up(l, t) if the first l terms are pairwise different and si = si+l = si+2l = ···= si+( t−1)l Downloaded 07/17/13 to 18.51.1.228. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php for every i,1≤ i ≤ l. For example, a, b, c, a, b, c, a, b, c, a, b, c Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 552 JACOB FOX, JANOS´ PACH, AND ANDREW SUK is a type up(3, 4) sequence or, in short, an up(3, 4) sequence. We need the following theorem of Klazar [13] on generalized Davenport–Schinzel sequences. Theorem 2.1 (Klazar). For l ≥ 2 and t ≥ 3, the length of any l-regular sequence over an n-element alphabet that does not contain a subsequence of type up(l, t) has length at most · (lt−3) · 10α(n)lt n l2 (10l) . ≥ For l 2, the sequence S = s1,s2,...,s3l−2 of length 3l − 2issaidtobeoftype up-down-up(l) if the first l terms are pairwise different and si = s2l−i = s(2l−2)+i ≤ ≤ for every i, 1 i l.
Load more

Recommended publications
  • Geometry © 2003 Springer-Verlag New York Inc
    Discrete Comput Geom 30:311–320 (2003) Discrete & Computational DOI: 10.1007/s00454-003-0012-9 Geometry © 2003 Springer-Verlag New York Inc. Unavoidable Configurations in Complete Topological Graphs∗ J´anos Pach,1,2 J´ozsef Solymosi,2,3 and G´eza T´oth2 1Courant Institute, New York University, New York, NY 10012, USA [email protected] 2R´enyi Institute, Hungarian Academy of Sciences, Pf 127, H-1364 Budapest, Hungary {pach, solymosi, geza}@renyi.hu 3Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4 Abstract. A topological graph is a graph drawn in the plane so that its vertices are represented by points, and its edges are represented by Jordan curves connecting the corre- sponding points, with the property that any two curves have at most one point in common. We define two canonical classes of topological complete graphs, and prove that every topo- logical complete graph with n vertices has a canonical subgraph of size at least c log1/8 n, which belongs to one of these classes. We also show that every complete topological graph with n vertices has a non-crossing subgraph isomorphic to any fixed tree with at most c log1/6 n vertices. 1. Introduction, Results A topological graph G is a graph drawn in the plane by Jordan curves, any two of which have at most one point in common. That is, it is defined as a pair (V (G), E(G)), where V (G) is a set of points in the plane and E(G) is a set of simple continuous arcs ∗ J´anos Pach was supported by NSF Grant CCR-00-98246 and PSC-CUNY Research Award 64421- 0034.
    [Show full text]
  • Arxiv:1909.00223V1 [Cs.CG] 31 Aug 2019
    Simple k-Planar Graphs are Simple (k + 1)-Quasiplanar∗ Patrizio Angeliniy Michael A. Bekosy Franz J. Brandenburgz Giordano Da Lozzox Giuseppe Di Battistax Walter Didimo{ Michael Hoffmannk Giuseppe Liotta{ Fabrizio Montecchiani{ Ignaz Rutterz Csaba D. T´oth∗∗ yy y Universit¨atT¨ubingen,T¨ubingen,Germany fangelini,[email protected] z University of Passau, Passau, Germany fbrandenb,[email protected] x Roma Tre University, Rome, Italy fgiordano.dalozzo,[email protected] { Universit´adegli Studi di Perugia, Perugia, Italy fwalter.didimo,giuseppe.liotta,[email protected] k Department of Computer Science, ETH Z¨urich, Z¨urich, Switzerland [email protected] ∗∗ California State University Northridge, Los Angeles, CA, USA [email protected] yy Tufts University, Medford, MA, USA Abstract A simple topological graph is k-quasiplanar (k ≥ 2) if it contains no k pairwise crossing edges, and k-planar if no edge is crossed more than k times. In this paper, we explore the relationship between k-planarity and k-quasiplanarity to show that, for k ≥ 2, every k-planar simple topological graph can be transformed into a (k + 1)-quasiplanar simple topological graph. arXiv:1909.00223v1 [cs.CG] 31 Aug 2019 ∗Preliminary versions of the results presented in this paper appeared at WG 2017 [6] and MFCS 2017 [19]. 1 (a) (b) (c) (d) Figure 1: (a) A crossing configuration that is forbidden in a 3-planar topological graph. (b) A 3-planar topological graph. (c) A crossing configuration that is forbidden in a 4-quasiplanar topological graph. (d) A 4-quasiplanar topological graph obtained from the one of Figure (b) by suitably rerouting the thick edge.
    [Show full text]
  • Mathematisches Forschungsinstitut Oberwolfach Combinatorics
    Mathematisches Forschungsinstitut Oberwolfach Report No. 1/2006 Combinatorics Organised by L´aszl´oLov´asz (Redmond) Hans J¨urgen Pr¨omel (Berlin) January 1st – January 7th, 2006 Abstract. This is the report on the Oberwolfach workshop on Combina- torics, held 1–7 January 2006. Combinatorics is a branch of mathematics studying families of mainly, but not exclusively, finite or countable struc- tures – discrete objects. The discrete objects considered in the workshop were graphs, set systems, discrete geometries, and matrices. The programme consisted of 15 invited lectures, 18 contributed talks, and a problem ses- sion focusing on recent developments in graph theory, coding theory, discrete geometry, extremal combinatorics, Ramsey theory, theoretical computer sci- ence, and probabilistic combinatorics. Mathematics Subject Classification (2000): 05Cxx, 05Dxx, 52Bxx, 68Rxx. Introduction by the Organisers The workshop Combinatorics organised by L´aszl´oLov´asz (Redmond) and Hans J¨urgen Pr¨omel (Berlin) was held January 1st–January 7th, 2006. This meeting was very well attended with 48 participants from many different countries. The pro- gramme consisted of 15 plenary lectures, accompanied by 18 shorter contributions and a vivid problem session led by Vera T. S´os. The plenary lectures provided a very good overview over the current develop- ments in several areas of combinatorics and discrete mathematics. We were very fortunate that some of the speaker reported on essential progress on longstanding open problems. In particular, Ajtai, Koml´os, Simonovits, and Szemer´edi solved the Erd˝os–T. S´os conjecture on trees (for large trees) and Tao and Vu greatly improved the asymptotic upper bound on the probability that a Bernoulli ma- trix is singular.
    [Show full text]
  • The Theory of Combinatorial Maps and Its Use in the Graph-Topological Computations
    The theory of combinatorial maps and its use in the graph-topological computations. Dainis Zeps To cite this version: Dainis Zeps. The theory of combinatorial maps and its use in the graph-topological computations.. Mathematics [math]. Institute of Mathematics and Computer Science. University of Latvia, 1998. English. tel-00417773 HAL Id: tel-00417773 https://tel.archives-ouvertes.fr/tel-00417773 Submitted on 18 Sep 2009 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. University of Latvia The theory of combinatorial maps and its use in the graph-topological computations Dr. Math. Dissertation Dainis Zeps Institute of Mathematics and Computer Science University of Latvia Rai»nabulv., 29, R¹³ga,LV-1459 Latvia R¹³ga,1997 Dainis ZEPS 2 The theory of combinatorial maps and its use in the graph-topological computations. Dainis ZEPS ¤ y ¤This research is supported by the grant 96.0247 of Latvian Council of Science. yAuthor's address: Institute of Mathematics and Computer Science, University of Latvia, 29 Rainis blvd., Riga, Latvia. [email protected] 1 Abstract In this work we investigate combinatorial maps, see [2, 11, 12, 18, 19, 22, 23], applying the ge- ometrical idea of considering the corners between the edges in the embedding of the graph on the surface to be the elements on which the permutations act [58].
    [Show full text]
  • Combinatorial Topology
    Topological Graph Theory 3 Basic Definitions 3 Some Common Graphs Kn 8 Connectivity 9 Mader's Theorem v 12 Menger's Theorem 15 Minors and Topological Minors Kn 19 Application: Reliability Polynomial 21 The Cycle Space 30 Inclusion-Exclusion The Dumbbell Graph 32 Some Applications to 3-connected Graphs 33 Tutte's Lemma A graph is a set of vertices 40 Betti Numbers 9 Trees which are connected by edges. Bottlenecks and Connectivity This is a simple example of Theorems and Lemmas 12 a topological space. Two 13 Mader's Theorem topological characteristics of 5 Average vertex degree 14 Dumbbell Graph graphs are the number of con- 6 An Even number of odd vertices 23 K5 Subdivision nected components, and the 16 Adding paths preserves ·(G) 2 26 Peterson Graph number of cycles they contain. ¸ 17 Menger's Theorem 27 Application: Reliability Polynomial 18 Characterization of 2-connected graphs 28 How to count cycles Homeomorphic Faces as Basis Definitions 24 47 29 Graphs a 1-complexes 51 Tutte Counterexample 1 Graph 36 Identifying Paths 52 Tutte's Lemma 4 Neighborhood 37 Identifying Cycles 7 Paths and Cycles 42 Definitions of Maps 11 Vertex Connectedness 44 Induced Map on Cycle Spaces 15 k-path connected 46 Inclusion-Exclusion 21 Graph Isomorphism 50 Tutte 22 Subdivision 53 Characterization of ·(G) 3 ¸ 25 Contraction Examples 30 Linearizeation of Sets 32 Summing the Target 2 Friends in the Order 33 Summing the Domain 3 Some Common Graphs 38 Cycle Space 8 Paths and Cycles Basic Definitions 1 A graph is a mathematical representation of network.
    [Show full text]
  • 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.
    [Show full text]
  • Disjoint Edges in Topological Graphs and the Tangled-Thrackle Conjecture
    Disjoint edges in topological graphs and the tangled-thrackle conjecture∗ Andres J. Ruiz-Vargas† Andrew Suk‡ Csaba D. T´oth§ September 22, 2018 Abstract It is shown that for a constant t N, every simple topological graph on n vertices has O(n) edges if the graph has no two sets of∈t edges such that every edge in one set is disjoint from all edges of the other set (i.e., the complement of the intersection graph of the edges is Kt,t-free). As an application, we settle the tangled-thrackle conjecture formulated by Pach, Radoiˇci´c, and T´oth: Every n-vertex graph drawn in the plane such that every pair of edges have precisely one point in common, where this point is either a common endpoint, a crossing, or a point of tangency, has at most O(n) edges. 1 Introduction A topological graph is a graph drawn in the plane such that its vertices are represented by distinct points and its edges are represented by Jordan arcs between the corresponding points satisfying the following (nondegeneracy) conditions: (a) no edge intersects any vertex other than its endpoints, (b) any two edges have only a finite number of interior points in common, (c) no three edges have a common interior point, and (d) if two edges share an interior point, then they properly cross at that point [7]. A topological graph is simple if every pair of edges intersect in at most one point. Two edges of a topological graph cross if their interiors share a point, and are disjoint if they neither share a common vertex nor cross.
    [Show full text]
  • A Review of Geometric, Topological and Graph Theory Apparatuses for the Modeling and Analysis of Biomolecular Data
    A review of geometric, topological and graph theory apparatuses for the modeling and analysis of biomolecular data Kelin Xia1 ∗and Guo-Wei Wei2;3 y 1Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371 2Department of Mathematics Michigan State University, MI 48824, USA 3Department of Biochemistry & Molecular Biology Michigan State University, MI 48824, USA January 31, 2017 Abstract Geometric, topological and graph theory modeling and analysis of biomolecules are of essential im- portance in the conceptualization of molecular structure, function, dynamics, and transport. On the one hand, geometric modeling provides molecular surface and structural representation, and offers the basis for molecular visualization, which is crucial for the understanding of molecular structure and interactions. On the other hand, it bridges the gap between molecular structural data and theoretical/mathematical mod- els. Topological analysis and modeling give rise to atomic critic points and connectivity, and shed light on the intrinsic topological invariants such as independent components (atoms), rings (pockets) and cavities. Graph theory analyzes biomolecular interactions and reveals biomolecular structure-function relationship. In this paper, we review certain geometric, topological and graph theory apparatuses for biomolecular data modeling and analysis. These apparatuses are categorized into discrete and continuous ones. For discrete approaches, graph theory, Gaussian network model, anisotropic network model, normal mode analysis, quasi-harmonic analysis, flexibility and rigidity index, molecular nonlinear dynamics, spectral graph the- ory, and persistent homology are discussed. For continuous mathematical tools, we present discrete to continuum mapping, high dimensional persistent homology, biomolecular geometric modeling, differential geometry theory of surfaces, curvature evaluation, variational derivation of minimal molecular surfaces, atoms in molecule theory and quantum chemical topology.
    [Show full text]
  • 10 GEOMETRIC GRAPH THEORY J´Anos Pach
    10 GEOMETRIC GRAPH THEORY J´anos Pach INTRODUCTION In the traditional areas of graph theory (Ramsey theory, extremal graph theory, random graphs, etc.), graphs are regarded as abstract binary relations. The relevant methods are often incapable of providing satisfactory answers to questions arising in geometric applications. Geometric graph theory focuses on combinatorial and geometric properties of graphs drawn in the plane by straight-line edges (or, more generally, by edges represented by simple Jordan arcs). It is a fairly new discipline abounding in open problems, but it has already yielded some striking results that have proved instrumental in the solution of several basic problems in combinatorial and computational geometry (including the k-set problem and metric questions discussed in Sections 1.1 and 1.2, respectively, of this Handbook). This chapter is partitioned into extremal problems (Section 10.1), crossing numbers (Section 10.2), and generalizations (Section 10.3). 10.1 EXTREMAL PROBLEMS Tur´an’s classical theorem [Tur54] determines the maximum number of edges that an abstract graph with n vertices can have without containing, as a subgraph, a complete graph with k vertices. In the spirit of this result, one can raise the follow- ing general question. Given a class of so-called forbidden geometric subgraphs, what is the maximum number of edgesH that a geometric graph of n vertices can have without containing a geometric subgraph belonging to ? Similarly, Ramsey’s the- orem [Ram30] for abstract graphs has some natural analoguesH for geometric graphs. In this section we will be concerned mainly with problems of these two types.
    [Show full text]
  • On Grids in Topological Graphs
    On Grids in Topological Graphs Eyal Ackerman∗ Jacob Fox† J´anos Pach‡ Andrew Suk§ Abstract A topological graph is a graph drawn in the plane with vertices represented by points and edges as arcs connecting its vertices. A k-grid in a topological graph is a pair of edge subsets, each of size k, such that every edge in one subset crosses every edge in the other subset. It is known that for a fixed constant k, every n-vertex topological graph with no k-grid has O(n) edges. We conjecture that this remains true even when: (1) considering grids with distinct vertices; or (2) all edges are straight-line segments and the edges within each subset of the grid are required to be pairwise ∗ 2 disjoint. These conjectures are shown to be true apart from log n and log n factors, respectively. We also settle the conjectures for some special cases, including the second conjecture for convex geometric graphs. This result follows from a stronger statement that generalizes the celebrated Marcus-Tardos Theorem on excluded patterns in 0-1 matrices. 1 Introduction The intersection graph of a set of geometric objects has the members of as its vertex set and an edge between every pair of objectsC with a nonempty intersection. The problemsC of finding maximum independent set and maximum clique in the intersection graph of geometric objects have received a considerable amount of attention in the literature due to their applications in VLSI design [HM85], map labeling [AKS98], frequency assignment in cellular networks [M97], and elsewhere. Here we study the intersection graph of the edge set of graphs that are drawn in the plane.
    [Show full text]
  • The Beginnings of Geometric Graph Theory
    The Beginnings of Geometric Graph Theory J´anosPach∗ \...to ask the right question and to ask it of the right person." (Richard Guy) Abstract Geometric graphs (topological graphs) are graphs drawn in the plane with possi- bly crossing straight-line edges (resp., curvilinear edges). Starting with a problem of Heinz Hopf and Erika Pannwitz from 1934 and a seminal paper of Paul Erd}osfrom 1946, we give a biased survey of Tur´an-type questions in the theory of geometric and topological graphs. What is the maximum number of edges that a geometric or topological graph of n vertices can have if it contains no forbidden subconfiguration of a certain type? We put special emphasis on open problems raised by Erd}osor directly motivated by his work. 1 Introduction The term \geometric graph theory" is often used to refer to a large, amorphous body of research related to graphs defined by geometric means. Here we take a narrower view: by a geometric graph we mean a graph G drawn in the plane with possibly intersecting straight-line edges. If the edges are allowed to be arbitrary continuous curves connecting the vertices (points), then G is called a topological graph. Disregarding the particular way the graph is drawn, we obtain the \abstract" underlying graph of G, which is usually also denoted by G. We use the term geometric graph theory as a short form for \the theory of geometric and topological graphs." In the past few decades, a number of exciting discoveries have been made in this field. Some of them have found interesting applications in graph drawing, in combina- torial and computational geometry, in additive number theory, and elsewhere.
    [Show full text]
  • Embedding Graphs Into Two-Dimensional Simplicial Complexes
    Embedding Graphs into Two-Dimensional Simplicial Complexes Éric Colin de Verdière1 Université Paris-Est, LIGM, CNRS, ENPC, ESIEE Paris, UPEM, Marne-la-Vallée France [email protected] Thomas Magnard2 Université Paris-Est, LIGM, CNRS, ENPC, ESIEE Paris, UPEM, Marne-la-Vallée France [email protected] Bojan Mohar3 Department of Mathematics, Simon Fraser University Burnaby, Canada [email protected] https://orcid.org/0000-0002-7408-6148 Abstract We consider the problem of deciding whether an input graph G admits a topological embedding into a two-dimensional simplicial complex C . This problem includes, among others, the embed- dability problem of a graph on a surface and the topological crossing number of a graph, but is more general. The problem is NP-complete when C is part of the input, and we give a polynomial-time algorithm if the complex C is fixed. Our strategy is to reduce the problem to an embedding extension problem on a surface, which has the following form: Given a subgraph H0 of a graph G0, and an embedding of H0 on a surface S, can that embedding be extended to an embedding of G0 on S? Such problems can be solved, in turn, using a key component in Mohar’s algorithm to decide the embeddability of a graph on a fixed surface (STOC 1996, SIAM J. Discr. Math. 1999). 2012 ACM Subject Classification Theory of computation → Computational geometry, Mathe- matics of computing → Graph algorithms, Mathematics of computing → Graphs and surfaces, Mathematics of computing → Algebraic topology Keywords and phrases computational topology, embedding, simplicial complex, graph, surface Digital Object Identifier 10.4230/LIPIcs.SoCG.2018.27 Related Version A full version of this paper is available at https://arxiv.org/abs/1803.
    [Show full text]