Genetic Algorithms and Book Embeddings: a Dual Layered Approach
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BOOK EMBEDDINGS of LAMAN GRAPHS 1. Definitions Definition 1
BOOK EMBEDDINGS OF LAMAN GRAPHS MICHELLE DELCOURT, MEGHAN GALIARDI, AND JORDAN TIRRELL Abstract. We explore various techniques to embed planar Laman graphs in books. These graphs can be embedded as pointed pseudo-triangulations and have many interesting properties. We conjecture that all planar Laman graphs can be embedded in two pages. 1. Definitions Definition 1. A graph G with n vertices and m edges is a Laman graph if m = 2n − 3 and every subgraph induced by k vertices (where k > 1) contains at most 2k − 3 edges. Definition 2. A book embedding of a graph G is a drawing with no crossings of G on a book; every edge is mapped to a page (a half-plane), and every vertex is mapped to the spine (the boundary shared by all the half-planes). Definition 3. The book thickness of a graph G, denoted bt(G) also known as page number, is the minimum number of pages needed to book embed G. 2. Finding Book Thickness Using Henneberg Constructions Motivated by Schnyder labelings for triangulations, Huemer and Kappes proposed an analogous binary labeling for quadrangulations which decom- poses the edge set in two directed trees rooted at a sink. A weak version of this labeling is valid for Laman graphs; however, the labeling does not guarantee a decomposition into two trees. The binary labeling of Laman graphs of Huemer and Kappes is based on Henneberg constructions. The Henneberg construction provides a decomposition into two trees whereas the labeling does not. A Henneberg construction consists of two moves: I. Adding a vertex of degree two II. -
Open Problems on Graph Drawing
Open problems on Graph Drawing Alexandros Angelopoulos Corelab E.C.E - N.T.U.A. February 17, 2014 Outline Introduction - Motivation - Discussion Variants of thicknesses Thickness Geometric thickness Book thickness Bounds Complexity Related problems & future work 2/ Motivation: Air Traffic Management Separation -VerticalVertical - Lateral - Longitudinal 3/ Motivation: Air Traffic Management : Maximization of \free flight” airspace c d c d X f f i1 i4 i0 i2 i3 i5 e e Y a b a b 8 Direct-to flight (as a choice among \free flight") increases the complexity of air traffic patterns Actually... 4 Direct-to flight increases the complexity of air traffic patterns and we have something to study... 4/ Motivation: Air Traffic Management 5/ How to model? { Graph drawing & thicknesses Geometric thickness (θ¯) Book thickness (bt) Dillencourt et al. (2000) Bernhart and Kainen (1979) : only straight lines : convex positioning of nodes v4 v5 v1 v2 θ(G) ≤ θ¯(G) ≤ bt(G) v5 v3 v0 v4 4 v1 Applications in VLSI & graph visualizationv3 Thickness (θ) v0 8 θ, θ¯, bt characterize the graph (minimizations over all allowedv2 drawings) Tutte (1963), \classical" planar decomposition 6/ Geometric graphs and graph drawings Definition 1.1 (Geometric graph, Bose et al. (2006), many Erd¨ospapers). A geometric graph G is a pair (V (G);E(G)) where V (G) is a set of points in the plane in general position and E(G) is set of closed segments with endpoints in V (G). Elements of V (G) are vertices and elements of E(G) are edges, so we can associate this straight-line drawing with the underlying abstract graph G(V; E). -
The 4-Girth-Thickness of the Complete Graph
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 14 (2018) 319–327 The 4-girth-thickness of the complete graph Christian Rubio-Montiel UMI LAFMIA 3175 CNRS at CINVESTAV-IPN, 07300, Mexico City, Mexico Division´ de Matematicas´ e Ingenier´ıa, FES Ac-UNAM, 53150, Naulcalpan, Mexico Received 10 March 2017, accepted 21 June 2017, published online 19 September 2017 Abstract In this paper, we define the 4-girth-thickness θ(4;G) of a graph G as the minimum number of planar subgraphs of girth at least 4 whose union is G. We prove that the 4- girth-thickness of an arbitrary complete graph K , θ(4;K ), is n+2 for n = 6; 10 and n n 4 6 θ(4;K6) = 3. Keywords: Thickness, planar decomposition, girth, complete graph. Math. Subj. Class.: 05C10 1 Introduction A finite graph G is planar if it can be embedded in the plane without any two of its edges g crossing. A planar graph of order n and girth g has size at most g 2 (n 2) (see [6]), and an acyclic graph of order n has size at most n 1, in this case,− we define− its girth as . The thickness θ(G) of a graph G is the minimum− number of planar subgraphs whose union1 is G; i.e. the minimum number of planar subgraphs into which the edges of G can be partitioned. The thickness was introduced by Tutte [11] in 1963. Since then, exact results have been obtained when G is a complete graph [1,3,4], a complete multipartite graph [5, 12, 13] or a hypercube [9]. -
Local Page Numbers
Local Page Numbers Bachelor Thesis of Laura Merker At the Department of Informatics Institute of Theoretical Informatics Reviewers: Prof. Dr. Dorothea Wagner Prof. Dr. Peter Sanders Advisor: Dr. Torsten Ueckerdt Time Period: May 22, 2018 – September 21, 2018 KIT – University of the State of Baden-Wuerttemberg and National Laboratory of the Helmholtz Association www.kit.edu Statement of Authorship I hereby declare that this document has been composed by myself and describes my own work, unless otherwise acknowledged in the text. I declare that I have observed the Satzung des KIT zur Sicherung guter wissenschaftlicher Praxis, as amended. Ich versichere wahrheitsgemäß, die Arbeit selbstständig verfasst, alle benutzten Hilfsmittel vollständig und genau angegeben und alles kenntlich gemacht zu haben, was aus Arbeiten anderer unverändert oder mit Abänderungen entnommen wurde, sowie die Satzung des KIT zur Sicherung guter wissenschaftlicher Praxis in der jeweils gültigen Fassung beachtet zu haben. Karlsruhe, September 21, 2018 iii Abstract A k-local book embedding consists of a linear ordering of the vertices of a graph and a partition of its edges into sets of edges, called pages, such that any two edges on the same page do not cross and every vertex has incident edges on at most k pages. Here, two edges cross if their endpoints alternate in the linear ordering. The local page number pl(G) of a graph G is the minimum k such that there exists a k-local book embedding for G. Given a graph G and a vertex ordering, we prove that it is NP-complete to decide whether there exists a k-local book embedding for G with respect to the given vertex ordering for any fixed k ≥ 3. -
DISSERTATION GENERALIZED BOOK EMBEDDINGS Submitted By
DISSERTATION GENERALIZED BOOK EMBEDDINGS Submitted by Shannon Brod Overbay Department of Mathematics In partial fulfillment of the requirements for the degree of Doctor of Philosophy Colorado State University Fort Collins, Colorado Summer 1998 COLORADO STATE UNIVERSITY May 13, 1998 WE HEREBY RECOMMEND THAT THE DISSERTATION PREPARED UNDER OUR SUPERVISION BY SHANNON BROD OVERBAY ENTITLED GENERALIZED BOOK EMBEDDINGS BE ACCEPTED AS FULFILLING IN PART REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSO- PHY. Committee on Graduate Work Adviser Department Head ii ABSTRACT OF DISSERTATION GENERALIZED BOOK EMBEDDINGS An n-book is formed by joining n distinct half-planes, called pages, together at a line in 3-space, called the spine. The book thickness bt(G) of a graph G is the smallest number of pages needed to embed G in a book so that the vertices lie on the spine and each edge lies on a single page in such a way that no two edges cross each other or the spine. In the first chapter, we provide background material on book embeddings of graphs and preview our results on several related problems. In the second chapter, we use a theorem of Bernhart and Kainen and a result of Whitney to present a large class of two-page embeddable planar graphs. In particular, we prove that a graph G that can be drawn in the plane so that G has no triangles other than faces can be embedded in a two-page book. The discussion of planar graphs continues in the third chapter where we define a book with a tree-spine. -
On the Book Thickness of $ K $-Trees
On the book thickness of k-trees∗ Vida Dujmovi´c† David R. Wood‡ May 29, 2018 Abstract Every k-tree has book thickness at most k + 1, and this bound is best possible for all k ≥ 3. Vandenbussche et al. (2009) proved that every k-tree that has a smooth degree-3 tree decomposition with width k has book thickness at most k. We prove this result is best possible for k ≥ 4, by constructing a k-tree with book thickness k + 1 that has a smooth degree-4 tree decomposition with width k. This solves an open problem of Vandenbussche et al. (2009) 1 Introduction Consider a drawing of a graph1 G in which the vertices are represented by distinct points on a circle in the plane, and each edge is a chord of the circle between the corresponding points. Suppose that each edge is assigned one of k colours such that crossing edges receive distinct colours. This structure is called a k-page book embedding of G: one can also think of the vertices as being ordered along the spine of a book, and the edges that receive the same colour being drawn on a single page of the book without crossings. The book thickness of G, denoted by bt(G), is the minimum integer k for which there is a k-page book embedding of G. Book embeddings, first defined by Ollmann [9], are ubiquitous structures with a vari- ety of applications; see [5] for a survey with over 50 references. A book embedding is also called a stack layout, and book thickness is also called stacknumber, pagenumber and fixed arXiv:0911.4162v1 [math.CO] 21 Nov 2009 outerthickness. -
[Cs.CG] 9 Sep 2002 ⋆⋆ Uhpors a Enmd;Se[,3 O Noeve.O H Ot P Imple More the Heuristic Effectiveness
On Simultaneous Graph Embedding C. A. Duncan1, A. Efrat∗2, C. Erten∗2, S. G. Kobourov⋆2, and J. S. B. Mitchell⋆⋆3 1 Dept. of Computer Science, Univ. of Miami, [email protected] 2 Dept. of Computer Science, Univ. of Arizona, {alon,erten,kobourov}@cs.arizona.edu 3 Dept. of Applied Mathematics and Statistics, Stony Brook University, [email protected] Abstract. We consider the problem of simultaneous embedding of pla- nar graphs. There are two variants of this problem, one in which the mapping between the vertices of the two graphs is given and another where the mapping is not given. In particular, we show that without mapping, any number of outerplanar graphs can be embedded simulta- neously on an O(n) × O(n) grid, and an outerplanar and general planar graph can be embedded simultaneously on an O(n2) × O(n3) grid. If the mapping is given, we show how to embed two paths on an n × n grid, or two caterpillar graphs on an O(n2) × O(n3) grid. 1 Introduction The areas of graph drawing and information visualization have seen significant growth in recent years. Often the visualization problems involve taking infor- mation in the form of graphs and displaying them in a manner that both is aesthetically pleasing and conveys some meaning. The aesthetic criteria by itself are the topic of much debate and research, but some generally accepted and tested standards include preferences for straight-line edges or those with only a few bends, a limited number of crossings, good separation of vertices and edges, as well as a small overall area. -
The Book Thickness of a Graph There Are Several Geometric
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 27, 320-331 (1979) The Book Thickness of a Graph FRANK BERNHART* Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada AND PAUL C. KAINEN+ Department of Mathematics and Statistics, Case Western Reserve University, Cleveland, Ohio Communicated by the Editors Received July 19, 1975 The book thickness bt(G) of a graph G is defined, its basic properties are delineated, and relations are given with other invariants such as thickness, genus, and chromatic number. A graph G has book thickness bt(G) < 2 if and only if it is a subgraph of a hamiltonian planar graph, but we conjecture that there are planar graphs with arbitrarily high book thickness. 1. INTRODUCTION There are several geometric invariants which have been studied extensively for graphs-among them, genus and thickness. In this paper we introduce a new invariant defined by considering embeddings of graphs into the members of what seems to us to be a very natural class of objects. For n >, 0, an n-book, or a book with n pages, consists of a line L in 3- space (called the spine) together with n distinct half-planes (called pages) with L as their common boundary. We usually adopt the convention that L is the z axis (in the standard parameterization of euclidean 3-space) and so is oriented “up” and “down.” An n-book embedding is a topological embedding /3 of G in an n-book which carries each vertex into the spine and each edge into the interior of at most one page. -
Graph Treewidth and Geometric Thickness Parameters
Graph Treewidth and Geometric Thickness Parameters Vida Dujmovi´c 1? and David R. Wood 2?? 1 School of Computer Science, Carleton University, Ottawa, Canada ([email protected]) 2 Departament de Matem`atica Aplicada II, Universitat Polit`ecnicade Catalunya, Barcelona, Spain ([email protected]) Abstract. Consider a drawing of a graph G in the plane such that cross- ing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness θ(G). By restricting the edges to be straight, we obtain the geometric thick- ness θ(G). By further restricting the vertices to be in convex position, we obtain the book thickness bt(G). This paper studies the relationship between these parameters and the treewidth of G. Let θ(Tk)/ θ(Tk) / bt(Tk) denote the maximum thickness / geometric thickness / book thickness of a graph with treewidth at most k. We prove that: – θ(Tk) = θ(Tk) = dk/2e, and – bt(Tk) = k for k ≤ 2, and bt(Tk) = k + 1 for k ≥ 3. The first result says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. The second result disproves the conjecture of Ganley and Heath [Discrete Appl. Math. 2001] that bt(Tk) = k for all k. Analogous results are proved for outerthickness, arboricity, and star-arboricity. 1 Introduction Partitions of the edge set of a graph G into a small number of ‘nice’ subgraphs is in the mainstream of graph theory. For example, in a proper edge colouring, the subgraphs of the partition are matchings. -
Planar Graphs, Biplanar Graphs and Graph Thickness
California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of aduateGr Studies 12-2016 PLANAR GRAPHS, BIPLANAR GRAPHS AND GRAPH THICKNESS Sean M. Hearon California State University - San Bernardino Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd Part of the Discrete Mathematics and Combinatorics Commons Recommended Citation Hearon, Sean M., "PLANAR GRAPHS, BIPLANAR GRAPHS AND GRAPH THICKNESS" (2016). Electronic Theses, Projects, and Dissertations. 427. https://scholarworks.lib.csusb.edu/etd/427 This Thesis is brought to you for free and open access by the Office of aduateGr Studies at CSUSB ScholarWorks. It has been accepted for inclusion in Electronic Theses, Projects, and Dissertations by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. Planar Graphs, Biplanar Graphs and Graph Thickness A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Sean Michael Hearon December 2016 Planar Graphs, Biplanar Graphs and Graph Thickness A Thesis Presented to the Faculty of California State University, San Bernardino by Sean Michael Hearon December 2016 Approved by: Dr. Jeremy Aikin, Committee Chair Date Dr. Cory Johnson, Committee Member Dr. Rolland Trapp, Committee Member Dr. Charles Stanton, Chair, Dr. Cory Dunn Department of Mathematics Graduate Coordinator, Department of Mathematics iii Abstract A graph is planar if it can be drawn on a piece of paper such that no two edges cross. The smallest complete and complete bipartite graphs that are not planar are K5 and K3;3. -
Upward Partitioned Book Embeddings
Upward Partitioned Book Embeddings The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Akitaya, Hugo A. et al. "Upward Partitioned Book Embeddings." International Symposium on Graph Drawing and Network Visualization, September 2018, Barcelona, Spain, Springer International Publishing, January 2018. © 2018 Springer International Publishing AG As Published http://dx.doi.org/10.1007/978-3-319-73915-1_18 Publisher Springer International Publishing Version Original manuscript Citable link https://hdl.handle.net/1721.1/128437 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ Upward Partitioned Book Embeddings Hugo A. Akitaya1, Erik D. Demaine2, Adam Hesterberg2, and Quanquan C. Liu2 1 Tufts University, Medford, MA, USA hugo.alves [email protected] 2 Massachusetts Institute of Technology, Cambridge, MA, USA edemaine,achester,[email protected] Abstract. We analyze a directed variation of the book embedding prob- lem when the page partition is prespecified and the nodes on the spine must be in topological order (upward book embedding). Given a directed acyclic graph and a partition of its edges into k pages, can we linearly or- der the vertices such that the drawing is upward (a topological sort) and each page avoids crossings? We prove that the problem is NP-complete for k ≥ 3, and for k ≥ 4 even in the special case when each page is a matching. By contrast, the problem can be solved in linear time for k = 2 pages when pages are restricted to matchings. The problem comes from Jack Edmonds (1997), motivated as a generalization of the map folding problem from computational origami. -
Arxiv:1908.08911V1 [Cs.DS] 23 Aug 2019
Parameterized Algorithms for Book Embedding Problems? Sujoy Bhore1, Robert Ganian1, Fabrizio Montecchiani2, and Martin N¨ollenburg1 1 Algorithms and Complexity Group, TU Wien, Vienna, Austria fsujoy,rganian,[email protected] 2 Engineering Department, University of Perugia, Perugia, Italy [email protected] Abstract. A k-page book embedding of a graph G draws the vertices of G on a line and the edges on k half-planes (called pages) bounded by this line, such that no two edges on the same page cross. We study the problem of determining whether G admits a k-page book embedding both when the linear order of the vertices is fixed, called Fixed-Order Book Thickness, or not fixed, called Book Thickness. Both problems are known to be NP-complete in general. We show that Fixed-Order Book Thickness and Book Thickness are fixed-parameter tractable parameterized by the vertex cover number of the graph and that Fixed- Order Book Thickness is fixed-parameter tractable parameterized by the pathwidth of the vertex order. 1 Introduction A k-page book embedding of a graph G is a drawing that maps the vertices of G to distinct points on a line, called spine, and each edge to a simple curve drawn inside one of k half-planes bounded by the spine, called pages, such that no two edges on the same page cross [20,25]; see Fig.1 for an illustration. This kind of layout can be alternatively defined in combinatorial terms as follows. A k-page book embedding of G is a linear order ≺ of its vertices and a coloring of its edges which guarantee that no two edges uv, wx of the same color have their vertices ordered as u ≺ w ≺ v ≺ x.