Algorithms and Combinatorics for Geometric Graphs Lecture Notes
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Crossing Numbers of Graphs with Rotation Systems
Crossing Numbers of Graphs with Rotation Systems Michael J. Pelsmajer∗ Marcus Schaefer Department of Applied Mathematics Department of Computer Science Illinois Institute of Technology DePaul University Chicago, Illinois 60616, USA Chicago, Illinois 60604, USA [email protected] [email protected] Daniel Štefankovič Computer Science Department University of Rochester Rochester, NY 14627-0226 [email protected] July 1, 2009 Abstract We show that computing the crossing number and the odd crossing number of a graph with a given rotation system is NP-complete. As a consequence we can show that many of the well-known crossing number notions are NP-complete even if restricted to cubic graphs (with or without rotation system). In particular, we can show that Tutte’s independent odd crossing number is NP-complete, and we obtain a new and simpler proof of Hliněný’s result that computing the crossing number of a cubic graph is NP-complete. We also consider the special case of multigraphs with rotation sys- tems on a fixed number k of vertices. For k =1 we give an O(m log m) algorithm, where m is the number of edges, and for loopless multi- graphs on 2 vertices we present a linear time 2-approximation algo- rithm. In both cases there are interesting connections to edit-distance problems on (cyclic) strings. For larger k we show how to approximate k+4 k the crossing number to within a factor of 4 /5 in time O(m log m) on a graph with m edges. ∗ Partially supported by NSA Grant H98230-08-1-0043. -
Surface Separation
Surface Separation Joseph Slote and Rob Thompson Abstract How many ways can an orientable surface be cut into k pieces such that every point on the cut is necessary? More formally, how many graphs have an embedding on the genus g torus such that 1) the surface is partitioned into k components, and 2) any subset of the embedding does not partition the g torus into k components? We show that answering this question is equivalent to the task of counting graphs that admit rotation systems satisfying two inequalities: one depending on the number of faces in the rotation system's cellular embedding, and one depending on the chromatic number of the cellular embedding's dual graph. Introduction Suppose we desire to cut a surface in such a way as to separate it into a fixed number of pieces with no extra or unnecessary cutting. More precisely, for a given surface S, we are interested in determining which multigraphs G admit embeddings T : G,! S such that 1. S n im(T ) has k connected components, and 2. For any proper subgraph G0 ⊂ G, S n T (G0) has fewer than k connected components. Definition 1. An embedding of a graph G into a surface S which satisfies the conditions 1 and 2 above will be called a minimal k-cut of S. (If only condition 1 is satisfied, we will call it a k-cut of S). A simple example helps illustrate the conditions for being a minimal k-cut. Example 2. The graph G = B2, a bouquet of two circles, admits a minimal 2-cut embedding on the torus, shown in Figure 0.1. -
Some Results on Convex Greedy Embedding Conjecture for 3
Some Results On Convex Greedy Embedding Conjecture for 3-Connected Planar Graphs Subhas Kumar Ghosh and Koushik Sinha∗ October 30, 2018 Abstract A greedy embedding of a graph G = (V,E) into a metric space (X,d) is a function x : V (G) X such that in the embedding for every pair of non-adjacent → vertices x(s),x(t) there exists another vertex x(u) adjacent to x(s) which is closer to x(t) than x(s). This notion of greedy embedding was defined by Papadimitriou and Ratajczak (Theor. Comput. Sci. 2005), where authors conjectured that every 3-connected planar graph has a greedy embedding (possibly planar and convex) in the Euclidean plane. Recently, greedy embedding conjecture has been proved by Leighton and Moitra (FOCS 2008). However, their algorithm do not result in a drawing that is planar and convex for all 3-connected planar graph in the Euclidean plane. In this work we consider the planar convex greedy embedding conjecture and make some progress. We derive a new characterization of planar convex greedy embedding that given a 3-connected planar graph G = (V,E), 2 an embedding x : V R of G is a planar convex greedy embedding if and → only if, in the embedding x, weight of the maximum weight spanning tree (T ) and weight of the minimum weight spanning tree (MST) satisfies wt(T )/wt(MST) 1−δ ≤ ( V 1) , for some 0 < δ 1. In order to present this result we define | |− ≤ a notion of weak greedy embedding. For β 1 a β–weak greedy embedding ≥ of a graph is a planar embedding x : V (G) X such that for every pair of → non-adjacent vertices x(s),x(t) there exists a vertex x(u) adjacent to x(s) such that distance between x(u) and x(t) is at most β times the distance between x(s) and x(t). -
Partial Duality of Hypermaps
PARTIAL DUALITY OF HYPERMAPS by S. Chmutov & F. Vignes-Tourneret Abstract.— We introduce partial duality of hypermaps, which include the classical Euler-Poincaré duality as a particular case. Combinatorially, hypermaps may be described in one of three ways: as three involutions on the set of flags (bi-rotation system or τ-model), or as three permutations on the set of half-edges (rotation system or σ-model in orientable case), or as edge 3-coloured graphs. We express partial duality in each of these models. We give a formula for the genus change under partial duality. MSC.— 05C10, 05C65, 57M15, 57Q15 Keywords and phrases.— maps, hypermaps, partial duality, permutational models, rotation system, bi-rotation system, edge coloured graphs Contents Introduction.............................................................. 2 1. Hypermaps............................................................... 3 2. Partial duality............................................................ 10 arXiv:1409.0632v2 [math.CO] 9 Feb 2021 3. Genus change............................................................. 17 4. Directions of future research.............................................. 19 2 S. CHMUTOV & F. VIGNES-TOURNERET Introduction Maps can be thought of as graphs embedded into surfaces. Hypermaps are hypergraphs embedded into surfaces. In other words, in hypermaps a (hyper) edge is allowed to connect more than two vertices, so having more than two half-edges, or just a single half-edge (see Figure1). One way of combinatorially study oriented hypermaps, the rotation system or the σ-model, is to consider permutations of its half-edges, also know as darts, around each vertex, around each hyperedge, and around each face according to the orientation. This model has been carefully worked out by R. Cori[Cor75], however it can be traced back to L. Heffter[Hef91]. -
Classification of Minimal Separating Sets in Low Genus Surfaces
Classification of Minimal Separating Sets in Low Genus Surfaces J. J. P. Veerman,∗ William Maxwell,† Victor Rielly,‡ Austin K. Williams§ July 1, 2021 Abstract Consider a surface S and let M ⊂ S. If S \ M is not connected, then we say M separates S, and we refer to M as a separating set of S. If M separates S, and no proper subset of M separates S, then we say M is a minimal separating set of S. In this paper we use computational methods of combinatorial topology to classify the minimal separating sets of the orientable surfaces of genus g = 2 and g = 3. The classification for genus 0 and 1 was done in earlier work, using methods of algebraic topology. Keywords: Combinatorial Topology, Graph Embeddings, Minimal Separating Sets, Rotation Systems. Surfaces and Separating Sets We begin with some basic definitions. Definition 1. When we say surface we are referring to a closed, connected, triangulated, orientable, 2-manifold. Let S be a surface and let M ⊂ S. Definition 2. If S \ M is not connected, then we say M separates S, and we refer to M as a separating set of S. Definition 3. If M separates S, and no proper subset of M separates S, then we say M is a minimal separating set of S. In this paper we will not consider pathological separating sets, such as the common boundary of the Lakes of Wada. To that end, when we refer to subsets of a surface it will be understood that we are referring to finitely triangulated subsets. -
Embedding Simply Connected 2-Complexes in 3-Space II
Embedding simply connected 2-complexes in 3-space II. Rotation systems Johannes Carmesin University of Birmingham September 5, 2019 Abstract We prove that 2-dimensional simplicial complexes whose first ho- mology group is trivial have topological embeddings in 3-space if and only if there are embeddings of their link graphs in the plane that are compatible at the edges and they are simply connected. 1 Introduction This is the second paper in a series of five papers. In the first paper [3] of this series we give an overview about this series as a whole. In this paper we give combinatorial characterisations for when certain simplicial complexes embed in 3-space. This completes the proof of a 3-dimensional analogue of Kuratowski's characterisation of planarity for graphs, started in [3]. A (2-dimensional) simplicial complex has a topological embedding in 3- space if and only if it has a piece-wise linear embedding if and only if it has a differential embedding [1, 6, 11].1 Perelman proved that every compact arXiv:1709.04643v3 [math.CO] 4 Sep 2019 3 simply connected 3-dimensional manifold is isomorphic to the 3-sphere S [13, 14, 15]. In this paper we use Perelman's theorem to prove a combina- torial characterisation of which simply connected simplicial complexes can 3 be topologically embedded into S as follows. The link graph at a vertex v of a simplicial complex is the graph whose vertices are the edges incident with v and whose edges are the faces incident 1 v e Figure 1: The link graph at the vertex v is indicated in grey. -
Realizing Graphs As Polyhedra
Realizing Graphs as Polyhedra David Eppstein Recent Trends in Graph Drawing: Curves, Graphs, and Intersections California State University Northridge, September 2015 Outline I Convex polyhedra I Steinitz's theorem I Incremental realization I Lifting from planar drawings I Circle packing I Non-convex polyhedra I Four-dimensional polytopes Steinitz's theorem Graphs of 3d convex polyhedra = 3-vertex-connected planar graphs [Steinitz 1922] File:Uniform polyhedron-53-t0.svg and File:Graph of 20-fullerene w-nodes.svg from Wikimedia commons Polyhedral graphs are planar Projection from near one face gives a planar Schlegel diagram Alternatively, the polyhedral surface itself (minus one point) is topologically equivalent to a plane Polyhedral graphs are 3-connected Theorem [Balinski 1961]: d-dimensional polytopes are d-connected Proof idea: I Consider any set C of fewer than d vertices I Add one more vertex v I Find linear function f , zero on C [ fvg, nonzero elsewhere I Simplex method finds C-avoiding paths from v to both min f and max f , and from all other vertices to min or max I Therefore, C does not disconnect the graph Combinatorial uniqueness of realization Face cycles do not separate rest of graph: Any two points not on given face can be connected by a path disjoint from the face (same argument as Balinski) Non-face cycles do separate one side of cycle from the other (Jordan curve theorem) So faces are determined by graph structure Planar graphs that are not 3-connected can have multiple embeddings Polyhedral realization of 3-connected -
Dessin D'enfant
RED DOCTORAL EN MATEMÁTICA -REGIÓN VALPARAÍSO Doctorado en Ciencias mención Matemática Líneas de Investigación: Algebra y Geometría (Superficies de Riemann, Grupos • Kleinianos, Geometría Algebraica) Estadística y Probabilidades • Análisis (Sistemas Dinámicos, Optimización, EDP, Análisis • Numérico) 2011 2 / 14 c RHO UTFSM LXXX ENCUENTRO ANUAL Ruben A. Hidalgo SOCIEDAD DE MATEMÁTICA DE CHILE UTFSM Termas El Corazón - Los Andes Región de Valparaíso 2011 1 3, 4 y 5 de Noviembre del 2011 Ruben A. Hidalgo UTFSM 2011 2 Ruben A. Hidalgo UTFSM 2011 3 Ruben A. Hidalgo UTFSM 2011 4 Ruben A. Hidalgo UTFSM 2011 5 Ruben A. Hidalgo UTFSM 2011 6 Los cálculos hechos en estas notas fueron, en gran parte, con la ayuda del sofware GAP. The GAP Group, GAP -- Groups, Algorithms, and Programming, Version 4.4.12; 2008. (http://www.gap-system.org) Ruben A. Hidalgo UTFSM 2011 7 Ruben A. Hidalgo UTFSM 2011 8 Ruben A. Hidalgo UTFSM 2011 9 Ruben A. Hidalgo UTFSM 2011 10 Ruben A. Hidalgo UTFSM 2011 11 Ruben A. Hidalgo UTFSM 2011 12 Ruben A. Hidalgo UTFSM 2011 13 Ruben A. Hidalgo UTFSM 2011 14 Ruben A. Hidalgo UTFSM 2011 15 Ruben A. Hidalgo UTFSM 2011 16 + Ruben A. Hidalgo UTFSM 2011 17 7 Ruben A. Hidalgo UTFSM 2011 18 6 8 x3 y4 6 8 x3 y4 Ruben A. Hidalgo UTFSM 2011 19 Ruben A. Hidalgo UTFSM 2011 20 belyi (Sage) Ruben A. Hidalgo7/21/11 11:10 AM UTFSM The Sage Notebook Searching for Sage server... 2011 21 Ruben A. Hidalgo admin Toggle Home Published Log Settings Report a Problem Help Sign out Version 4.2 UTFSM belyi Save Save & quit Discard & quit last edited on July2011 21, 2011 10:42 AM by admin File.. -
Planar and Toroidal Morphs Made Easier
Planar and Toroidal Morphs Made Easier Jeff Erickson Patrick Lin University of Illinois Urbana-Champaign Abstract We present simpler algorithms for two closely related morphing problems, both based on the barycentric interpolation paradigm introduced by Floater and Gotsman, which is in turn based on Floater’s asymmetric extension of Tutte’s classical spring-embedding theorem. First, we give a very simple algorithm to construct piecewise-linear morphs between planar straight-line graphs. Specifically, given isomorphic straight-line drawings Γ0 and Γ1 of the same 3-connected planar graph G, with the same convex outer face, we construct a 1+!=2 morph from Γ0 to Γ1 that consists of O(n) unidirectional morphing steps, in O(n ) time. Our algorithm entirely avoids the classical edge-collapsing strategy dating back to Cairns; instead, in each morphing step, we interpolate the pair of weights associated with a single edge. Second, we describe a natural extension of barycentric interpolation to geodesic graphs on the flat torus. Barycentric interpolation cannot be applied directly in this setting, because the linear systems defining intermediate vertex positions are not necessarily solvable. We describe a simple scaling strategy that circumvents this issue. Computing the appropriate !=2 scaling requires O(n ) time, after which we can can compute the drawing at any point in !=2 the morph in O(n ) time. Our algorithm is considerably simpler than the recent algorithm of Chambers et al. and produces more natural morphs. Our techniques also yield a simple proof of a conjecture of Connelly et al. for geodesic torus triangulations. 1 Introduction Computing morphs between geometric objects is a fundamental problem that has been well studied, with many applications in graphics, animation, modeling, and more. -
The Topological Theory of Current Graphs This Paper
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF COMBINATORIAL THEORY (B) 17, 218-233 (1974) The Topological Theory of Current Graphs JONATHAN L. GROSS* Department of Mathematical Statistics, Columbia University, New York, New York 10027 AND SETH R. ALPERT~ Department of Medical Computer Science, SUNY Downstate Medical Center, Brooklyn, New York 11203 Communicated by Alan J. Hoffin Received April 17, 1973 W. Gustin’s introduction of combinatorial current graphs as a device for obtaining orientable imbeddings of Cayley “color” graphs was fundamental to the solution of the Heawood map-coloring problem by G. Ringel, J. W. T. Youngs, C. M. Terry, and L. R. Welch. The topological current graphs of this paper lead to a construction that generalizes the method of Gustin and its augmentation to “vortex” graphs by Youngs, extending the scope of current graph theory from Cayley graphs alone to the much larger class of graphs that are covering spaces. 1. INTR~DuOD~N This paper provides proof of our previously announced result [83 that any orientable imbedding of a Cayley graph derived from a combinatorial current graph is a (possibly branched) covering of the dual of the implicit imbedding of that combinatorial current graph. This result is a special case of our Theorem 2 for graph imbeddings derived from what we call “topological current graphs”. We intend to describe our topological methods in language as close as possible to that of G. Ringel and J. W. T. Youngs, using elementary examples to illustrate such structures as * The fist author did most of this work as an IBM Postdoctoral Fellow in Mathe- matical Sciences at the IBM Thomas J. -
A Graph-Based Formalism for Surface Codes and Twists
A graph-based formalism for surface codes and twists Rahul Sarkar∗ and Theodore J. Yoder† January 26, 2021 Abstract Twist defects in surface codes can be used to encode more logical qubits, improve the code rate, and implement logical gates. In this work we provide a rigorous formalism for constructing surface codes with twists generalizing the well-defined homological formalism introduced by Kitaev for describing CSS surface codes. In particular, we associate a surface code to any graph G embedded on any 2D-manifold, in such a way that (1) qubits are associated to the vertices of the graph, (2) stabilizers are associated to faces, (3) twist defects are associated to odd-degree vertices. In this way, we are able to reproduce the variety of surface codes, with and without twists, in the literature and produce some new examples. We also calculate and bound various code properties such as the rate and distance in terms of topological graph properties such as genus, systole, and face-width. Contents 1 Introduction 3 2 Preliminaries 4 2.1 Graph embeddings . .5 2.2 Oriented rotation systems . .5 2.3 General rotation systems . .6 2.4 Dual graph and checkerboardability . .7 2.5 Homology of graph embeddings . .9 2.6 Covering maps and contractible loops . 10 3 Majorana and qubit surface codes from rotation systems 11 arXiv:2101.09349v1 [quant-ph] 22 Jan 2021 3.1 Majorana operators and codes . 11 3.2 Majorana surface codes defined on embedded graphs . 13 3.3 From Majorana surface codes to qubit surface codes . 16 3.4 Relation to homological surface codes . -
Embeddings of Polytopes and Polyhedral Complexes
University of California Los Angeles Embeddings of Polytopes and Polyhedral Complexes A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Stedman McDee Wilson 2012 c Copyright by Stedman McDee Wilson 2012 Abstract of the Dissertation Embeddings of Polytopes and Polyhedral Complexes by Stedman McDee Wilson Doctor of Philosophy in Mathematics University of California, Los Angeles, 2012 Professor Igor Pak, Chair When does a topological polyhedral complex (embedded in Rd) admit a geometric realiza- tion (a rectilinear embedding in Rd)? What are the constraints on the possible realizations? Two classic results concerning such questions are F´ary'stheorem, which states that every planar graph can be drawn in the plane such that each edge is a straight line segment, and Tutte's theorem, which provides necessary and sufficient conditions for embedding a planar graph such that all faces are convex. The present work is motivated largely by the question of whether these types of results generalize to higher dimensions. We begin by constructing an irrational polytopal complex consisting of 1278 convex 3-polytopes in R3. The methods of this construction may also be used to produce small topological complexes with no geometric realization, as well as geometric complexes which encode arbitrary point and line configurations. This allows us to prove universality theorems for 3-dimensional polytopal complexes and arrangements. We also investigate geometric realizations of plane triangulations, and describe an explicit algorithm that embeds a plane triangulation with n vertices in a 4n3 × 8n5 integer grid, in such a way that at each step of the algorithm, the resulting region of the plane is convex.