DOCSLIB.ORG

MA Algorithms for Crossing Minimization in Book Drawings

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Algorithms for crossing minimisation in book drawings Master Thesis of Jonathan Klawitter At the Department of Informatics Institute of Theoretical Informatics Chair of Algorithmics I Reviewer: Prof. Dr. Dorothea Wagner Second reviewer: Priv.-Doz. Dr. Martin Nöllenburg Advisor: Dr. Tamara Mchedlidze Duration:: 23. October 2015 – 22. April 2016 KIT – University of the State of Baden-Wuerttemberg and National Research Center of the Helmholtz Association www.kit.edu I declare that I have developed and written the enclosed thesis completely by myself, and have not used sources or means without declaration in the text. Karlsruhe, 20th April 2016 ......................................... (Jonathan Klawitter) Abstract A book with k pages consists of a line (the spine) and k half-planes (the pages ), each with the spine as boundary. In a k-page book drawing of a graph the vertices lie on the spine, and each edge is drawn as arc in one page. The minimal number of edge crossings in a k-page book drawing of a graph is called its k-page crossing number, which, in general, is -hard to determine [BE14]. A book drawing can be described by an order of the NP vertices on the spine and a distribution of the edges to pages. To compute book drawings, we combine vertex order heuristics with edge distribution heuristics and create heuristics that compute both simultaneously. We experimentally evaluate the performance of these heuristics on a new test suite that is centred on di↵erent graph classes. It turns out that our new heuristics produce drawings with fewer crossings for most of the tested graphs. We further investigate the optimisation of book drawings with force-based approaches, greedy and evolutionary algorithms as well as the successful combinations of the latter two. Zusammenfassung Ein Buch (book )mitk Seiten besteht aus einer Geraden (dem Bund (spine)) und k Hal- bebenen (den Seiten (pages )), die jeweils den Bund als Grenze haben. In einer k-seitigen Buchzeichnung (k-page book drawing) eines Graphen wird jeder Knoten auf den Bund und jede Kante als Kreisbogen auf eine Seite gezeichnet. Die minimale Anzahl von Kan- tenkreuzungen in einer k-seitigen Buchzeichnung eines Graphen nennt man seine k-Seiten Kreuzungszahl (k-page crossing number), welche im Allgemeinen -schwer zu bestimmen NP ist [BE14]. Eine Buchzeichnung kann durch die Ordnung der Knoten auf dem Bund und die Verteilung der Kanten auf die Seiten beschrieben werden. Um eine Buchzeichnung zu erzeugen, kombinieren wir Heuristiken zum Bestimmen von Knotenordnungen und Kan- tenverteilungen sowie neue Heuristiken, die beides simultan berechnen. Wir evaluieren die Performanz dieser Heuristiken anhand einer neuen Testsammlung, welches auf ver- schiedenen Graphklassen basiert. Wir konnten zeigen, dass unsere neuen Heuristiken fur¨ die meisten der getesteten Graphen Zeichnungen mit weniger Kreuzungen erzeugen. Des Weiteren betrachten wir die Optimierung von Graphzeichnungen mit kr¨aftebasierten und evolution¨aren Ans¨atzen, sowie Greedyalgorithmen und vielversprechende Kombinationen selbiger. v Contents 1. Introduction 1 1.1. Book drawings and book embeddings . 1 1.2. Motivation . 3 1.3. Related work . 4 1.4. Outline ...................................... 6 2. The problem 7 2.1. Basicresults.................................... 7 2.1.1. Trees.................................... 7 2.1.2. Biconnected components . 10 2.1.3. Outerplanar graphs . 10 2.1.4. Fixed vertex order . 11 2.2. NP-hardness . 12 2.2.1. Book embedding . 12 2.2.2. Crossing number . 13 2.3. Exact solution . 14 2.3.1. SAT.................................... 14 2.3.2. Weighted MAX-SAT . 16 2.4. Counting crossings . 17 2.4.1. Counting and reporting . 17 2.4.2. Open edges sweep . 17 2.4.3. Transformation to two-layer cross counting . 18 2.4.4. Evaluation . 19 3. Heuristic approaches 23 3.1. Vertexorderheuristics . 23 3.1.1. Depth-first search . 24 3.1.2. Breath-first search . 24 3.1.3. Max-neighboring . 25 3.1.4. Connectivity . 25 3.2. Edgedistributionheuristics . 27 3.2.1. Greedy .................................. 27 3.2.2. Slope . 28 3.2.3. Ear decomposition . 29 3.3. Fulldrawingheuristics . 30 4. Evaluation of heuristics 33 4.1. Linear number of edges . 34 4.1.1. Experimentalresults . 34 4.2. Outerplanar graphs . 36 4.3. Cartesian product of cycles . 37 4.3.1. Experimentalresults . 38 vii viii Contents 4.4. Planar graphs . 39 4.4.1. Finding Hamiltonian cycle . 40 4.4.2. Distributingperfectly . 41 4.4.3. Average number of crossings . 41 4.5. 1-planar graphs . 44 4.5.1. Experimentalresults . 44 4.6. K-trees....................................... 46 4.6.1. Experimentalresults . 46 4.7. Hypercubes and t-ary d-cubes . 48 4.7.1. Experimentalresults . 48 4.8. Complete graphs . 49 4.9. Complete bipartite graphs . 51 4.9.1. Experimentalresults . 51 4.10. Quadratic number of edges . 52 4.11. Conclusion . 54 5. Optimisation 57 5.1. Greedy optimisation . 57 5.1.1. Greedy vertex order optimisation . 57 5.1.2. Greedy edge distribution optimisation . 58 5.1.3. Greedy book drawing optimisation . 58 5.1.4. Evaluation . 59 5.2. Evolutionary algorithms . 60 5.2.1. Mimicing evolution in nature . 60 5.2.2. Implementation . 62 5.2.3. Algorithms . 64 5.3. Evaluation of optimisation algorithms . 65 5.4. Force-based optimisation . 67 5.4.1. Mean & median iteration vertex order . 67 5.4.2. Force-based edge distribution . 69 5.5. Further approaches . 71 5.5.1. Graph partitioning & clustering . 71 5.5.2. Maximal independent set . 72 6. Summary and outlook 73 Bibliography 76 Appendix 89 A. Graph generation . 89 A.1. Trees.................................... 89 A.2. Outerplanar graphs . 89 A.3. Maximal planar graphs . 89 A.4. Hamiltonian planar graphs . 90 A.5. 1-planar graphs . 90 A.6. K-trees .................................. 91 A.7. Random graphs . 91 A.8. Shu✏ing.................................. 91 B. Further experiment results . 91 B.1. 1-planar graphs . 91 B.2. Hypercubes and t-ary d-cubes . 92 B.3. Random graphs . 92 C. Toolsandmachines................................ 93 viii 1. Introduction In this thesis we evaluate heuristics and optimisation algorithms for the NP-hard problem of crossing minimisation in book drawings. We start with basic definitions and then proceed with our research question and related work. At the end of the chapter we give an overview of this thesis. 1.1. Book drawings and book embeddings We draw undirected graphs into topological structures fittingly called books. They can be defined as follows. Definition 1.1. For k 1,ak-page book,orabookwithk pages, consists of a line in ≥ three dimensional space (the spine) and k half-planes (the pages) bounded by this line. Less formal, a book consist solely of a spine with attached pages. A graph is drawn into a book by representing its vertices as distinct points on the spine and drawing each edge as a circular arc in exactly one page. In Figure 1.1 a graph with eight vertices is drawn into a book with four pages. page 1 page 3 1 8 spine page 2 page 4 Figure 1.1.: A 4-page book drawing of a graph, where vertices (from 1 to 8) are placed on the spine (shown twice) and edges are distributed to pages. Pages 1 and 2 are crossing-free, whereas the edges in page 3 and 4 produce several crossings. We note that we can turn around the book, meaning reverse the order of the vertices, and still have a proper book drawing. Furthermore, the numbering of the pages has no further relevance. The drawing stays combinatorially and topologically equivalent. Another feature of book drawings becomes apparent, if we transform them into circular drawings by bending the spine into a cycle, as shown in Figure 1.2. Here, vertices are 1 2 1. Introduction placed on a circle and edges are drawn as chords. If we cut the circle at a di↵erent position than where we joined the ends of the spine, we get a new drawing with rotated order of the vertices. This is also illustrated in Figure 1.2. To represent pages in circular drawings the edges are assigned to layers or are drawn in di↵erent colours. Figure 1.3 depicts the 4-page book from Figure 1.1 as circular drawing. Throughout this thesis, we will call the order (respectively circular order) of the vertices, vertex order, and the assignment of edges to pages, edge distribution. Joining these insights, we can now define the book drawings from the combinatorial point of view. Figure 1.2.: The 1-page book drawing on the left is converted into a circular drawing. The latter is then cut between two other vertices to obtain a book drawing with rotated spine. 1 8 page 1 page 2 page 3 page 4 Figure 1.3.: A circular drawing of the book drawing in Figure 1.1, where the pages are represented by the colours of the edges. Definition 1.2. A k-page book drawing of a graph G =(V,E) is defined by a vertex order, a circular order of its vertices V ,andanedgedistribution,amapofitsedgesE to k pages. Similar to planar drawings, there can also be crossings of edges in book drawings. We say two edges uv and xy cross if their arcs (respectively chords) cross. See for example again Figures 1.1 and 1.3 where pages 3 and 4 contain crossings. These are in fact the same crossings, since the transformation described above preserves them. We further observe that a crossing is only possible between non-incident edges and if their end vertices alternate. Hence, we get the following definition for crossings in book drawings. Definition 1.3. In a book drawing of a graph G =(V,E),twoofitsedgesuv, xy E 2 cross if they are on the same page and their endpoints alternate in the vertex order.
Recommended publications
  • K-Outerplanar Graphs, Planar Duality, and Low Stretch Spanning Trees

    K-Outerplanar Graphs, Planar Duality, and Low Stretch Spanning Trees

    k-Outerplanar Graphs, Planar Duality, and Low Stretch Spanning Trees Yuval Emek∗ Abstract Low distortion probabilistic embedding of graphs into approximating trees is an extensively studied topic. Of particular interest is the case where the approximating trees are required to be (subgraph) spanning trees of the given graph (or multigraph), in which case, the focus is usually on the equivalent problem of finding a (single) tree with low average stretch. Among the classes of graphs that received special attention in this context are k-outerplanar graphs (for a fixed k): Chekuri, Gupta, Newman, Rabinovich, and Sinclair show that every k-outerplanar graph can be probabilistically embedded into approximating trees with constant distortion regardless of the size of the graph. The approximating trees in the technique of Chekuri et al. are not necessarily spanning trees, though. In this paper it is shown that every k-outerplanar multigraph admits a spanning tree with constant average stretch. This immediately translates to a constant bound on the distortion of probabilistically embedding k-outerplanar graphs into their spanning trees. Moreover, an efficient randomized algorithm is presented for constructing such a low average stretch spanning tree. This algorithm relies on some new insights regarding the connection between low average stretch spanning trees and planar duality. Keywords: planar graphs, outerplanarity, average stretch, planar dual. ∗Microsoft Israel R&D Center, Herzelia, Israel and School of Electrical Engineering, Tel Aviv University, Tel Aviv, Israel. E-mail: [email protected]. Supported in part by the Israel Science Foundation, grants 221/07 and 664/05. 1 Introduction The problem.
  • Domination in Circle Graphs

    Domination in Circle Graphs

    Circles graphs Dominating set Some positive results Open Problems Domination in circle graphs Nicolas Bousquet Daniel Gon¸calves George B. Mertzios Christophe Paul Ignasi Sau St´ephanThomass´e WG 2012 Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems 1 Circles graphs 2 Dominating set 3 Some positive results 4 Open Problems Domination in circle graphs W [1]-hardness Under some algorithmic hypothesis, the W [1]-hard problems do not admit FPT algorithms. Circles graphs Dominating set Some positive results Open Problems Parameterized complexity FPT A problem parameterized by k is FPT (Fixed Parameter Tractable) iff it admits an algorithm which runs in time Poly(n) · f (k) for any instances of size n and of parameter k. Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Parameterized complexity FPT A problem parameterized by k is FPT (Fixed Parameter Tractable) iff it admits an algorithm which runs in time Poly(n) · f (k) for any instances of size n and of parameter k. W [1]-hardness Under some algorithmic hypothesis, the W [1]-hard problems do not admit FPT algorithms. Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Circle graphs Circle graph A circle graph is a graph which can be represented as an intersection graph of chords in a circle. 3 3 4 2 5 4 2 6 1 1 5 7 7 6 Domination in circle graphs Independent dominating sets. Connected dominating sets. Total dominating sets. All these problems are NP-complete. Circles graphs Dominating set Some positive results Open Problems Dominating set 3 4 5 2 6 1 7 Dominating set Set of chords which intersects all the chords of the graph.
  • Characterizations of Restricted Pairs of Planar Graphs Allowing Simultaneous Embedding with Fixed Edges

    Characterizations of Restricted Pairs of Planar Graphs Allowing Simultaneous Embedding with Fixed Edges

    Characterizations of Restricted Pairs of Planar Graphs Allowing Simultaneous Embedding with Fixed Edges J. Joseph Fowler1, Michael J¨unger2, Stephen Kobourov1, and Michael Schulz2 1 University of Arizona, USA {jfowler,kobourov}@cs.arizona.edu ⋆ 2 University of Cologne, Germany {mjuenger,schulz}@informatik.uni-koeln.de ⋆⋆ Abstract. A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the Euclidean plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same simple curve in the simultaneous drawing. Determining in polynomial time which pairs of graphs share a simultaneous embedding with fixed edges (SEFE) has been open. We give a necessary and sufficient condition for when a pair of graphs whose union is homeomorphic to K5 or K3,3 can have an SEFE. This allows us to determine which (outer)planar graphs always an SEFE with any other (outer)planar graphs. In both cases, we provide efficient al- gorithms to compute the simultaneous drawings. Finally, we provide an linear-time decision algorithm for deciding whether a pair of biconnected outerplanar graphs has an SEFE. 1 Introduction In many practical applications including the visualization of large graphs and very-large-scale integration (VLSI) of circuits on the same chip, edge crossings are undesirable. A single vertex set can be used with multiple edge sets that each correspond to different edge colors or circuit layers. While the pairwise union of all edge sets may be nonplanar, a planar drawing of each layer may be possible, as crossings between edges of distinct edge sets are permitted.
  • Minor-Closed Graph Classes with Bounded Layered Pathwidth

    Minor-Closed Graph Classes with Bounded Layered Pathwidth

    Minor-Closed Graph Classes with Bounded Layered Pathwidth Vida Dujmovi´c z David Eppstein y Gwena¨elJoret x Pat Morin ∗ David R. Wood { 19th October 2018; revised 4th June 2020 Abstract We prove that a minor-closed class of graphs has bounded layered pathwidth if and only if some apex-forest is not in the class. This generalises a theorem of Robertson and Seymour, which says that a minor-closed class of graphs has bounded pathwidth if and only if some forest is not in the class. 1 Introduction Pathwidth and treewidth are graph parameters that respectively measure how similar a given graph is to a path or a tree. These parameters are of fundamental importance in structural graph theory, especially in Roberston and Seymour's graph minors series. They also have numerous applications in algorithmic graph theory. Indeed, many NP-complete problems are solvable in polynomial time on graphs of bounded treewidth [23]. Recently, Dujmovi´c,Morin, and Wood [19] introduced the notion of layered treewidth. Loosely speaking, a graph has bounded layered treewidth if it has a tree decomposition and a layering such that each bag of the tree decomposition contains a bounded number of vertices in each layer (defined formally below). This definition is interesting since several natural graph classes, such as planar graphs, that have unbounded treewidth have bounded layered treewidth. Bannister, Devanny, Dujmovi´c,Eppstein, and Wood [1] introduced layered pathwidth, which is analogous to layered treewidth where the tree decomposition is arXiv:1810.08314v2 [math.CO] 4 Jun 2020 required to be a path decomposition.
  • Coloring Clean and K4-Free Circle Graphs 401

    Coloring Clean and K4-Free Circle Graphs 401

    Coloring Clean and K4-FreeCircleGraphs Alexandr V. Kostochka and Kevin G. Milans Abstract A circle graph is the intersection graph of chords drawn in a circle. The best-known general upper bound on the chromatic number of circle graphs with clique number k is 50 · 2k. We prove a stronger bound of 2k − 1 for graphs in a simpler subclass of circle graphs, called clean graphs. Based on this result, we prove that the chromatic number of every circle graph with clique number at most 3 is at most 38. 1 Introduction Recall that the chromatic number of a graph G, denoted χ(G), is the minimum size of a partition of V(G) into independent sets. A clique is a set of pairwise adjacent vertices, and the clique number of G, denoted ω(G), is the maximum size of a clique in G. Vertices in a clique must receive distinct colors, so χ(G) ≥ ω(G) for every graph G. In general, χ(G) cannot be bounded above by any function of ω(G). Indeed, there are triangle-free graphs with arbitrarily large chromatic number [4, 18]. When graphs have additional structure, it may be possible to bound the chromatic number in terms of the clique number. A family of graphs G is χ-bounded if there is a function f such that χ(G) ≤ f (ω(G)) for each G ∈G. Some families of intersection graphs of geometric objects have been shown to be χ-bounded (see e.g. [8,11,12]). Recall that the intersection graph of a family of sets has a vertex A.V.
  • BOOK EMBEDDINGS of LAMAN GRAPHS 1. Definitions Definition 1

    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

    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).
  • NP-Completeness of List Coloring and Precoloring Extension on the Edges of Planar Graphs

    NP-Completeness of List Coloring and Precoloring Extension on the Edges of Planar Graphs

    NP-completeness of list coloring and precoloring extension on the edges of planar graphs D´aniel Marx∗ 17th October 2004 Abstract In the edge precoloring extension problem we are given a graph with some of the edges having a preassigned color and it has to be decided whether this coloring can be extended to a proper k-edge-coloring of the graph. In list edge coloring every edge has a list of admissible colors, and the question is whether there is a proper edge coloring where every edge receives a color from its list. We show that both problems are NP-complete on (a) planar 3-regular bipartite graphs, (b) bipartite outerplanar graphs, and (c) bipartite series-parallel graphs. This improves previous results of Easton and Parker [6], and Fiala [8]. 1 Introduction In graph vertex coloring we have to assign colors to the vertices such that neighboring vertices receive different colors. Starting with [7] and [28], a gener- alization of coloring was investigated: in the list coloring problem each vertex can receive a color only from its prescribed list of admissible colors. In the pre- coloring extension problem a subset W of the vertices have preassigned colors and we have to extend this precoloring to a proper coloring of the whole graph, using only colors from a given color set C. It can be viewed as a special case of list coloring: the list of a precolored vertex consists of a single color, while the list of every other vertex is C. A thorough survey on list coloring, precoloring extension, and list chromatic number can be found in [26, 1, 12, 13].
  • The 4-Girth-Thickness of the Complete Graph

    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

    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

    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

    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.