DOCSLIB.ORG

INF562, Lecture 3: Geometric and Combinatorial Properties of Planar Graphs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
INF562, Lecture 3: Geometric and Combinatorial Properties of Planar Graphs INF562, Lecture 3: Geometric and combinatorial properties of planar graphs mardi 22 janvier 2013 Luca Castelli Aleardi 1-1 Intro Graph drawing: motivations and applications 2-1 Graph drawing and data visualization Global transportation system 3-1 Graph drawing and data visualization Roads, railways, ... 4-1 Graph drawing and data visualization Social network graph 5-1 Planar graphs french roads network Design of integrated circuits (VLSI) 6-1 Planar graphs french roads network Design of integrated circuits (VLSI) www.2m40.com 9 accidents en 2012 (last one, on 28th sptember) 6-2 Meshes and graphs in computational geometry Delaunay triangulations, Voronoi diagrams, planar meshes, ... GIS Technology triangles meshes already used in early 19th century (Delambre et Mchain) Delaunay triangulation Terrain modelling Planar mesh by L. Rineau, M. Yvinec Spherical Parameterization (Sheffer Gotsman) Voronoi diagram 7-1 Mesh parameterization in geometry processing 8-1 Mesh parameterization in geometry processing 9-1 Mesh parameterization in geometry processing 10-1 Graph drawing: motivation 0 1 1 0 1 1 Challenge: what kind of graph does AG represent? 1 0 1 1 0 1 1 1 0 1 1 0 AG = 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 11-1 Graph drawing: motivation 0 1 1 0 1 1 Challenge: what kind of graph does AG represent? 1 0 1 1 0 1 1 1 0 1 1 0 AG = 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 11-2 Graph drawing: motivation 0 1 1 0 1 1 Challenge: what kind of graph does AG represent? 1 0 1 1 0 1 1 1 0 1 1 0 AG = 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 11-3 Graph drawing: motivation 0 1 1 0 1 1 Challenge: what kind of graph does AG represent? 1 0 1 1 0 1 1 1 0 1 1 0 AG = 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 11-4 Graph drawing: motivation 0 1 1 0 1 1 Challenge: what kind of graph does AG represent? 1 0 1 1 0 1 1 1 0 1 1 0 AG = 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 11-5 Graph drawing: motivation 0 1 1 0 1 1 Challenge: what kind of graph does AG represent? 1 0 1 1 0 1 1 1 0 1 1 0 AG = 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 11-6 Graph drawing: motivation 0 1 1 0 1 1 Challenge: what kind of graph does AG represent? 1 0 1 1 0 1 1 1 0 1 1 0 AG = 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 11-7 Graph drawing: motivation 0 1 1 0 1 1 Challenge: what kind of graph does AG represent? 1 0 1 1 0 1 1 1 0 1 1 0 AG = 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 11-8 Part I Major results in graph theory 12-1 Major results (on planar graphs) in graph theory 13-1 Major results (on planar graphs) in graph theory Kuratowski theorem (1930) (cfr Wagner’s theorem, 1937) • G contains neither K5 nor K3;3 as minors 13-2 Major results (on planar graphs) in graph theory Kuratowski theorem (1930) (cfr Wagner’s theorem, 1937) • G contains neither K5 nor K3;3 as minors F´arytheorem (1947) • Every (simple) planar graph admits a straight line planar embedding (no edge crossings) v1 v1 v1 v2 v2 v4 v4 v2 v3 v5 v3 v5 dessin non planaire de G un dessin planaire de G v3 13-3 Major results (on planar graphs) in graph theory Thm (Steinitz, 1916) F´arytheorem (1947) • Every (simple) planar graph admits a straight line planar embedding (no edge crossings) v1 v1 v1 v2 v2 v4 v4 v2 v3 v5 v3 v5 dessin non planaire de G un dessin planaire de G v3 13-4 Major results (on planar graphs) in graph theory Thm (Steinitz, 1916) 3-connected planar graphs are the 1- skeletons of convex polyhedra Thm (Whitney, 1933) 3-connected planar graphs admit a unique planar embedding (up to homeomorphism and inversion of the sphere). 13-5 Major results (on planar graphs) in graph theory Thm (Steinitz, 1916) 3-connected planar graphs are the 1- skeletons of convex polyhedra Thm (Whitney, 1933) 3-connected planar graphs admit a unique Def G is 3-connected if planar embedding (up to homeomorphism and is connected and inversion of the sphere). the removal of one or two vertices does not disconnect G at least 3 vertices are required to disconnect the graph 13-6 Major results (on planar graphs) in graph theory Thm (Koebe-Andreev-Thurston) Every planar graph with n vertices is isomorphic to the intersection graph of n disks in the plane. Thm (Colin de Verdi`ere,1990) Theorem (Lovasz Schrijver ’99) Given a 3-connected planar graph G, the eigenvectors ξ ; ξ ; ξ of a CdV Colin de Verdiere invariant (multiplicity of λ eigen- 2 3 4 2 matrix defines a convex polyhedron containing the origin.. value of a generalized laplacian) • µ(G) ≤ 3 λ1 = −4; λ2 = λ3 = λ4 = 0 v2 M ξx ξy ξz −1 −1 −1 −1 −1 −1 −1 −1 v0 −1 −1 −1 v3 −1 −1 −1 −1 v1 0 1 0 −1 −1 −1 −1 v2 0 0 1 v3 1 0 0 v0 v1 13-7 Major results (on planar graphs) in graph theory Thm (Tutte barycentric method, 1963) Every 3-connected planar graph G admits a barycentric representation ρ in R2. X P ρ(vi) = wijρ(vj) ( j wij = 1 and wij > 0 j2N(i) 2 ρ :(VG) −! R is barycentric iff for each inner node vi, ρ(vi) is the barycenter of Get a straight line drawing the images of its neighbors solving a system a linear equations N(v4) = fv1; v2; v3; v5g N(v5) = fv2; v3; v4g 3 −1 −1 −1 0 −1 4 −1 −1 −1 M · x = ax v1 v1 L = −1 −1 4 −1 −1 v2 f M · y = ay v4 −1 −1 −1 4 −1 0 −1 −1 −1 3 v2 v3 v5 laplacian matrix v3 13-8 Major results (on planar graphs) in graph theory Theorem (Schnyder ’89) A graph G is planar if and only if the di- mension of its incidence poset is at most 3 Theorem (Schnyder, Soda ’90) For a triangulation T having n vertices, we can draw it on a grid of size (2n − 5) × (2n − 5), by setting v0 = (2n − 5; 0), v1 = (0; 0) and v2 = (0; 2n − 5). v2 (a) (b) v5 v4 v3 v0 v1 v3 (1; 2; 4) v4 (2; 4; 1) v0 v1 v5 (4; 1; 2) 13-9 Part II What is a surface mesh? (a short digression on embedded graphs, simplicial complexes and topological and combinatorial maps) 14-1 What is a (surface) mesh? surface mesh: set of vertices, edges and faces (polygons) defining a polyhedral surface in embedded in 3D (discrete approximation of a shape) Combinatorial structure + geometric embedding ”Connectivity”: the underlying map vertex coordinates incidence relations between triangles, vertices and edges 15-1 Planar and surface meshes: definition planar triangulation planar triangulation planar map embedded in R3 spherical drawing straight line drawing of a dodecahedron triangle mesh spherical parameterizations of a triangle mesh (Gotsman, Gu Sheffer, 2003) toroidal map (Eric Colin de Verdiere)` 16-1 Surface meshes as simplicial complexes abstract simplicial complex K (set of simplices) V = fv0; v1; : : : ; vn−1g E = ffi; jg; fk; lg;:::g F = ffi; j; kg; fj; i; lg;:::g inclusion property: ρ 2 K and σ ⊂ ρ −! σ 2 K not valid simplicial complex intersection property: given two simplices σ1; σ2 of K, the intersection σ1 [ σ2 is a face of both 17-1 Surface meshes as (topological) maps (geomettric realizations of maps) two different embeddings of the same graph A graph G = (V; E) is a pair of: • a set of vertices V = (v1; : : : ; vn) • a collection of E = (e1; : : : ; em) elements of the cartesian product V × V = f(u; v) j u 2 V; v 2 V g (edges). cellular embeddings of a graph defining the same (planar) map un dessin planaire est un plongement cellulaire de G dans R2, qui satisfait les conditions suivantes: (i) les sommets du graphe sont reprsent´ es´ par des points ; (ii) les aretes sont reprsent´ ees´ par des arcs de courbes ne se coupant qu’aux sommets ; (iii) les faces sont simplement connexes. (topologica) map: cellular embedding up to homeomorphism (equivalence class) two cellular embeddings defining the same planar map 18-1 Surface meshes as combinatorial maps (geomettric realizations of maps) 17 22 23 18 3 permutations on the set H of the 2n half-edges 20 2 10 19 11 6 1 3 5 8 21 24 (i) α involution without fixed point; 14 4 12 15 (ii) ασφ = Id; 7 16 13 9 (iii) the goup generated by σ, α et φ transitively on H. φ = (1; 2; 3; 4)(17; 23; 18; 22)(5; 10; 8; 12)(21; 19; 24; 15) ::: α = (2; 18)(4; 7)(12; 13)(9; 15)(14; 16)(10; 23) ::: 19-1 Surface meshes as combinatorial maps (geomettric realizations of maps) 17 22 23 18 3 permutations on the set H of the 2n half-edges 20 2 10 19 11 6 1 3 5 8 21 24 (i) α involution without fixed point; 14 4 12 15 (ii) ασφ = Id; 7 16 13 9 (iii) the goup generated by σ, α et φ transitively on H.
Load more

Recommended publications
  • On Tutte's Paper "How to Draw a Graph"
    ON HOW TO DRAW A GRAPH JIM GEELEN Abstract. We give an overview of Tutte's paper, \How to draw a graph", that contains: (i) a proof that every simple 3-connected planar graph admits a straight-line embedding in the plane such that each face boundary is a convex polygon, (ii) an elegant algo- rithm for finding such an embedding, (iii) an algorithm for testing planarity, and (iv) a proof of Kuratowski's theorem. 1. Introduction We give an overview on Tutte's paper on \How to draw a graph". This exposition is based on Tutte's paper [1], as well as a treatment of the topic by Lov´asz[2], and a series of talks by Chris Godsil. A convex embedding of a circuit C is an injective function φ : V (C) ! R2 that maps C to a convex polygon with each vertex of C being a corner of the polygon. A convex embedding of a 2-connected planar graph G is an injective function φ : V ! R2 such that (i) φ determines a straight-line plane embedding of G, and (ii) each facial circuit of the embedding is convex. (The condition that G is 2-connected ensures that faces are bounded by circuits.) Tutte [1] proves that every simple 3-connected planar graph admits a convex embedding. He also gives an efficient and elegant algorithm for finding such an embedding, he gives and efficient planarity testing algorithm, and he proves Kuratowski's Theorem. There are faster pla- narity testing algorithms and shorter proofs of Kuratowski's Theorem, but the methods are very attractive.
    [Show full text]
  • How to Draw a Graph
    HOW TO DRAW A GRAPH By W. T. TUTTE [Received 22 May 1962] 1. Introduction WE use the definitions of (11). However, in deference to some recent attempts to unify the terminology of graph theory we replace the term 'circuit' by 'polygon', and 'degree' by 'valency'. A graph G is 3-connected (nodally 3-connected) if it is simple and non-separable and satisfies the following condition; if G is the union of two proper subgraphs H and K such that HnK consists solely of two vertices u and v, then one of H and K is a link-graph (arc-graph) with ends u and v. It should be noted that the union of two proper subgraphs H and K of G can be the whole of G only if each of H and K includes at least one edge or vertex not belonging to the other. In this paper we are concerned mainly with nodally 3-connected graphs, but a specialization to 3-connected graphs is made in § 12. In § 3 we discuss conditions for a nodally 3-connected graph to be planar, and in § 5 we discuss conditions for the existence of Kuratowski subgraphs of a given graph. In §§ 6-9 we show how to obtain a convex representation of a nodally 3-connected graph, without Kuratowski subgraphs, by solving a set of linear equations. Some extensions of these results to general graphs, with a proof of Kuratowski's theorem, are given in §§ 10-11. In § 12 we discuss the representation in the plane of a pair of dual graphs, and in § 13 we draw attention to some unsolved problems.
    [Show full text]
  • Structural Graph Theory Doccourse 2014: Lecture Notes
    Structural Graph Theory DocCourse 2014: Lecture Notes Andrew Goodall, Llu´ısVena (eds.) 1 Structural Graph Theory DocCourse 2014 Editors: Andrew Goodall, Llu´ısVena Published by IUUK-CE-ITI series in 2017 2 Preface The DocCourse \Structural Graph Theory" took place in the autumn semester of 2014 under the auspices of the Computer Science Institute (IUUK)´ and the Department of Applied Mathematics (KAM) of the Faculty of Mathematics and Physics (MFF) at Charles University, supported by CORES ERC-CZ LL- 1201 and by DIMATIA Prague. The schedule was organized by Prof. Jaroslav Neˇsetˇriland Dr Andrew Goodall, with the assistance of Dr Llu´ısVena. A web page was maintained by Andrew Goodall, which provided links to lectures slides and further references.1 The Structural Graph Theory DocCourse followed the tradition established by those of 2004, 2005 and 2006 in Combinatorics, Geometry, and Computation, organized by Jaroslav Neˇsetˇriland the late JiˇriMatouˇsek,and has itself been followed by a DocCourse in Ramsey Theory in autumn of last year, organized by Jaroslav Neˇsetˇriland Jan Hubiˇcka. For the Structural Graph Theory DocCourse in 2014, five distinguished vis- iting speakers each gave a short series of lectures at the faculty building at Malostransk´en´amest´ı25 in Mal´aStrana: Prof. Matt DeVos of Simon Fraser University, Vancouver; Prof. Johann Makowsky of Technion - Israel Institute of Technology, Haifa; Dr. G´abor Kun of ELTE, Budapest; Prof. Michael Pinsker of Technische Universit¨atWien/ Universit´eDiderot - Paris 7; and Dr Lenka Zdeborov´aof CEA & CNRS, Saclay. The audience included graduate students and postdocs in Mathematics or in Computer Science in Prague and a handful of students from other universities in the Czech Republic and abroad.
    [Show full text]
  • Robust Network Computation David Pritchard AUG 14 2006 LIBRARIES ARCHIVES
    Robust Network Computation by David Pritchard Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Master of Engineering in Computer Science at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2005 @ :Massachusetts Institute of Technology 2005. All rights reserved. Author...,.. .... Depart. .....ent of ...........Electrical Engineering............. and..... Computer............... Science August 16, 2005 tAA I Certified by ........................... Santosh S. Vempala Associate Professor of Applied Mathematics Thesis Supervisor •.--.•-.. / Accepted by .......................... ..I ... ............ Arthur C. Smith Chairman, Department Committee on Graduate Theses MASSACHUSIS INST E OF TECHNOLOGY ARCHIVES AUG 14 2006 LIBRARIES ):· Robust Network Computation by David Pritchard Submitted to the Department of Electrical Engineering and Computer Science on August 16, 2005, in partial fulfillment of the requirements for the degree of Master of Engineering in Computer Science Abstract In this thesis, we present various models of distributed computation and algorithms for these models. The underlying theme is to come up with fast algorithms that can tolerate faults in the underlying network. We begin with the classical message-passing model of computation, surveying many known results. We give a new, universally optimal, edge-biconnectivity algorithm for the classical model. We also give a near-optimal sub-linear algorithm for identifying bridges, when all nodes are activated simultaneously. After discussing some ways in which the classical model is unrealistic, we survey known tech- niques for adapting the classical model to the real world. We describe a new balancing model of computation. The intent is that algorithms in this model should be automatically fault-tolerant. Existing algorithms that can be expressed in this model are discussed, including ones for clustering, maximum flow, and synchronization.
    [Show full text]
  • Large Non-Planar Graphs and an Application to Crossing-Critical Graphs
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Combinatorial Theory, Series B 101 (2011) 111–121 Contents lists available at ScienceDirect Journal of Combinatorial Theory, Series B www.elsevier.com/locate/jctb Large non-planar graphs and an application to crossing-critical graphs Guoli Ding a,1,BogdanOporowskia, Robin Thomas b,2, Dirk Vertigan a,3 a Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA b School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA article info abstract Article history: We prove that, for every positive integer k, there is an integer Received 5 January 2010 N such that every 4-connected non-planar graph with at least N Available online 24 December 2010 vertices has a minor isomorphic to K4,k, the graph obtained from a cycle of length 2k + 1 by adding an edge joining every pair Keywords: of vertices at distance exactly k, or the graph obtained from a Non-planar graph Crossing number cycle of length k by adding two vertices adjacent to each other Crossing-critical and to every vertex on the cycle. We also prove a version of this Minor for subdivisions rather than minors, and relax the connectivity to Subdivision allow 3-cuts with one side planar and of bounded size. We deduce that for every integer k there are only finitely many 3-connected 2-crossing-critical graphs with no subdivision isomorphic to the graph obtained from a cycle of length 2k by joining all pairs of diagonally opposite vertices.
    [Show full text]
  • Infinite Circuits in Locally Finite Graphs
    Infinite circuits in locally finite graphs Dissertation zur Erlangung des Doktorgrades des Fachbereichs Mathematik der Universit¨at Hamburg vorgelegt von Henning Bruhn Hamburg 2005 Contents 1 Introduction 1 1.1 Cycles in finite graphs . 1 1.2 Infinite cycles . 2 1.3 A topological definition of circles . 3 1.4 The topological cycle space . 6 1.5 The identification topology . 7 1.6 Overview . 10 2 Gallai's theorem and faithful cycle covers 11 2.1 Introduction . 11 2.2 Proof of Gallai's theorem . 13 2.3 Faithful cycle covers . 15 3 MacLane's and Kelmans' planarity criteria 19 3.1 Introduction . 19 3.2 Discussion of MacLane's criterion . 20 3.3 Simple generating sets . 21 3.4 The backward implication . 27 3.5 The forward implication . 29 3.6 Kelmans' planarity criterion . 31 4 Duality 33 4.1 Introduction . 33 4.2 Duality in infinite graphs . 35 4.3 Proof of the duality theorem . 38 4.4 Locally finite duals . 42 4.5 Duality in terms of spanning trees . 43 4.6 Alternative proof of Theorem 4.20 . 46 4.7 Colouring-flow duality and circuit covers . 47 4.8 MacLane's criterion revisited . 48 i 5 End degrees and infinite circuits 51 5.1 Introduction . 51 5.2 Defining end degrees . 52 5.3 A cut criterion for the end degree . 56 5.4 Proof of Theorem 5.4 . 61 5.5 Properties of the end degree . 65 5.6 Weakly even ends . 70 6 Hamilton cycles and long cycles 73 6.1 Introduction . 73 6.2 Hamilton circuits and covers .
    [Show full text]
  • On Covers of Graphs by Cayley Graphs
    On covers of graphs by Cayley graphs Agelos Georgakopoulos∗ Mathematics Institute University of Warwick CV4 7AL, UK Abstract We prove that every vertex transitive, planar, 1-ended, graph covers every graph whose balls of radius r are isomorphic to the ball of radius r in G for a sufficiently large r. We ask whether this is a general property of finitely presented Cayley graphs, as well as further related questions. 1 Introduction We will say that a graph H is r-locally-G if every ball of radius r in H is isomorphic to the ball of radius r in G. The following problem arose from a discussion with Itai Benjamini, and also appears in [5]. Problem 1.1. Does every finitely presented Cayley graph G admit an r ∈ N such that G covers every r-locally-G graph? The condition of being finitely presented is important here: for example, no such r exists for the standard Cayley graph of the lamplighter group Z ≀ Z2. Benjamini & Ellis [4] show that r = 2 suffices for the square grid Z2, while r = 3 suffices for the d-dimensional lattice (i.e. the standard Cayley graph of Zd for any d ≥ 3. The main result of this paper is Theorem 1.1. Let G be a vertex transitive planar 1-ended graph. Then there is r ∈ N such that G covers every r-locally-G graph (normally). Here, we say that a cover c : V (G) → V (H) is normal, if for every v, w ∈ arXiv:1504.00173v1 [math.GR] 1 Apr 2015 V (G) such that c(v)= c(w), there is an automorphism α of G such that α(v)= α(w) and c ◦ α = c.
    [Show full text]
  • Group Connectivity and Group Coloring a Ph.D
    Downloaded from orbit.dtu.dk on: Oct 05, 2021 Group Connectivity and Group Coloring A Ph.D. thesis in Graph Theory Langhede, Rikke Marie Publication date: 2020 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Langhede, R. M. (2020). Group Connectivity and Group Coloring: A Ph.D. thesis in Graph Theory. Technical University of Denmark. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Ph.D. Thesis Group Connectivity and Group Coloring A Ph.D. thesis in Graph Theory Rikke Marie Langhede Kongens Lyngby 2020 DTU Compute Department of Applied Mathematics and Computer Science Technical University of Denmark Matematiktorvet Building 303B 2800 Kongens Lyngby, Denmark Phone +45 4525 3031 [email protected] www.compute.dtu.dk Abstract This Ph.D. thesis in mathematics concerns group connectivity and group coloring within the general area of graph theory.
    [Show full text]
  • Immersion and Embedding of 2-Regular Digraphs
    Immersion and embedding of 2-regular digraphs Matt DeVos 1 Introduction In this section we will be interested in 2-regular digraphs (i.e. digraphs for which every vertex has both indegree and outdegree equal to 2). There is a natural operation called splitting which takes a 2-regular digraph and reduces to a new 2-regular digraph. To split a vertex v with inward edges uv and u0v and outward edges vw and vw0, we delete the vertex v and then add either the edges uw and u0w0, or we add the edges uw0 and u0w. If G; H are 2-regular digraphs, we say that H is immersed in G if a graph isomorphic to H may be obtained from G by a sequence of splits. u w u w u0 w0 v or u0 w0 u w u0 w0 Figure 1: splitting a vertex Our central goal in this article will be to show how the theory of 2-regular digraphs under immersion behaves similar to the theory of (undirected) graphs under graph minor operations. We will begin with some motivation. Consider an ordinary undirected graph G which is embedded in an orientable surface. The medial graph H is constructed from G by the following procedure. For every edge e of G add a vertex of the graph H in the centre of e. Now, whenever two edges e; f 2 E(G) are consecutive at a vertex (or equivalently, consecutive along a face) we add an edge between the corresponding vertices of H. Based on this construction, every vertex of the original graph is contained in the centre of a face of the medial graph, and every face of the original graph completely contains a new face of the 1 medial graph.
    [Show full text]
  • Iterated Altans and Their Properties
    MATCH MATCH Commun. Math. Comput. Chem. 74 (2015) 645-658 Communications in Mathematical and in Computer Chemistry ISSN 0340 - 6253 Iterated Altans and Their Properties Nino Baˇsi´c∗a, TomaˇzPisanskib aFaculty of Mathematics and Physics, University of Ljubljana, Slovenia, [email protected] bFAMNIT, University of Primorska, Slovenia, [email protected] (Received April 30, 2015) Abstract Recently a class of molecular graphs, called altans, became a focus of attention of several theoretical chemists and mathematicians. In this paper we study primary iterated altans and show, among other things, their connections with nanotubes and nanocaps. The question of classification of bipartite altans is also addressed. Using the results of Gutman we are able to enumerate Kekul´estructures of several nanocaps of arbitrary length. 1 Introduction Altans were first introduced as special planar systems, obtained from benzenoids by at- tachment of a ring to all outer vertices of valence two [19,26], in particular in connection with concentric decoupled nature of the ring currents (see papers by Zanasi et al. [25,26] and also Mallion and Dickens [8,9]). The graph-theoretical approach to ring current was initiated by Milan Randi´cin 1976 [31]. It was also studied by Gomes and Mallion [11]. Full description is provided for instance in [29, 30]. Moreover, see paper [10] by Patrick Fowler and Wendy Myrvold on `The Anthracene Problem'. ∗Corresponding author. -646- Later altans were generalized by Ivan Gutman [18] to arbitrary graphs. We essentially follow Gutman's approach. Our point of departure is a peripherally rooted graph, i.e. an ordered pair (G; S) which consists of an arbitrary graph G and a cyclically ordered subset S of its vertices, called the peripheral root.
    [Show full text]
  • Peripheral Circuits in Infinite Binary Matroids
    Peripheral circuits in infinite binary matroids N. Bowler, R. Christian, and R. B. Richter∗ University of Hamburg, Perth (Australia), and University of Waterloo LATEX-ed: June 28, 2018 Abstract A peripheral circuit in a (possibly infinite) matroid M is a circuit C of M such that M=C is connected. In the case of a 3-connected graph G, this is equivalent to C being chordless and G − V (C) being connected. Two classical theorems of Tutte assert that, for a 3-connected graph G: (i) every edge e of G is in two peripheral cycles that intersect just on e and its incident vertices; and (ii) the peripheral cycles generate the cycle space of G [12]. Bixby and Cunningham generalized these to binary matroids, with (i) requiring a small adaptation. Bruhn generalized (i) and (ii) to the Freudenthal compactification of a locally finite graph. We unify these two generalizations to “cofinitary, binary B-matroids". (Higgs introduced the B-matroid as an infinite matroid; recent works show this should now be accepted as the right notion of infinite matroid. Cofinitary means every cocircuit is finite.) Keywords: infinite matroid, binary matroid, peripheral circuits. AMS Classification (2000): 05B35, 05C63 1 Introduction A peripheral circuit in a (possibly infinite) matroid M is a circuit C of M such that M=C is connected. In the case of a 3-connected graph G, this is equivalent to C being chordless and G − V (C) being connected. Two classical theorems of Tutte [12] assert that, for a 3-connected graph G: (i) every edge e of G is in two peripheral cycles that intersect just on e and its incident vertices; and (ii) the peripheral cycles generate the cycle space of G.
    [Show full text]
  • Non-Planar Extensions of Subdivisions of Planar Graphs
    NON-PLANAR EXTENSIONS OF SUBDIVISIONS OF PLANAR GRAPHS Sergey Norin1 Department of Mathematics and Statistics McGill University Montreal, Quebec H3A 2K6, Canada and Robin Thomas2 School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332-0160, USA arXiv:1402.1999v3 [math.CO] 21 May 2019 5 November 1998, Revised 19 December 2018. Published in J. Combin. Theory Ser. B 121 (2016), 326-366. This version fixes an error in the published paper. The error was kindly pointed out to us by Katherine Naismith. Changes from the published version are indicated in red. 1 Supported by an NSERC discovery grant. 2 Partially supported by NSF under Grants No. DMS-9623031, DMS-0200595 and DMS-1202640, and by NSA under Grant No. MDA904-98-1-0517. ABSTRACT Almost 4-connectivity is a weakening of 4-connectivity which allows for vertices of degree three. In this paper we prove the following theorem. Let G be an almost 4-connected triangle-free planar graph, and let H be an almost 4-connected non-planar graph such that H has a subgraph isomorphic to a subdivision of G. Then there exists a graph G′ such that G′ is isomorphic to a minor of H, and either (i) G′ = G + uv for some vertices u, v ∈ V (G) such that no facial cycle of G contains both u and v, or ′ (ii) G = G + u1v1 + u2v2 for some distinct vertices u1,u2, v1, v2 ∈ V (G) such that u1,u2, v1, v2 appear on some facial cycle of G in the order listed. This is a lemma to be used in other papers.
    [Show full text]