The Geometric Thickness of Low Degree Graphs
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A Note on Simultaneous Embedding of Planar Graphs (Abstract)∗
EWCG 2005, Eindhoven, March 9–11, 2005 A Note on Simultaneous Embedding of Planar Graphs (Abstract)∗ Emilio Di Giacomo† Giuseppe Liotta† 1 Introduction 2 Preliminaries Let G1 and G2 be a pair of planar graphs such that The algorithms to compute a simultaneous embedding V (G1)=V (G2)=V .Asimultaneous embedding [6] of a pair of planar graphs and of a pair of trees pre- Ψ=(Γ1, Γ2)ofG1 and G2 is a pair of crossing-free sented in [6] are an elegant combination of the tech- drawings Γ1 and Γ2 of G1 and G2, respectively, such nique by Kaufmann and Wiese [7] for point-set em- ∈ that for every vertex v V we have Γ1(v)=Γ2(v). bedding and of the simultaneous embedding strategy ∈ ∩ If every edge e E(G1) E(G2) is represented with for two paths by Brass et al. [2]. In this section, we the same simple open Jordan curve both in Γ1 and in first recall the main ideas of Erten and Kobourov for Γ we say that Ψ is a simultaneous embedding with 2 simultaneous embedding of two planar graphs G1 and fixed edges. If the edges of G and G are repre- 1 2 G2 and then make a couple of observations on how to sented with straight-line segments in Γ1 and Γ2 we combine these ideas with known literature in order to say that Ψ is a simultaneous geometric embedding. extend some of the results in [6]. The existence of simultaneous geometric embeddings Suppose first that both G1 and G2 are Hamilto- for pairs of paths, cycles, and caterpillars is shown nian. -
Graph Drawing Dagstuhl Seminar
05191 Abstracts Collection Graph Drawing Dagstuhl Seminar Michael Jünger1, Stephen Kobourov2 and Petra Mutzel3 1 Univ. of Köln, DE 2 Univ. of Arizona, US [email protected] 3 Univ. of Dortmund, DE [email protected] Abstract. From 08.05. to 13.05.05, the Dagstuhl Seminar 05191 Graph Drawing was held in the International Conference and Research Cen- ter (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The rst section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available. Keywords. Graph Drawing, Visualization, Layout Algorithms, Inter- active Visualization 05191 Executive Summary Graph Drawing This paper summarizes the topics, aims, and achievements of the Dagstuhl Sem- inar 05191 on Graph Drawing. Keywords: Graph drawing Joint work of: Jünger, Michael; Kobourov, Stephen; Mutzel, Petra Full Paper: http://drops.dagstuhl.de/opus/volltexte/2006/342 05191 Open Problem Session Report This is a report on an informal session intended to stimulate communication and sharing of problems. Hence the attributions and citations may contain inaccu- racies, and are certainly not complete. Keywords: Open problems, graph drawing Joint work of: Whitesides, Sue Full Paper: http://drops.dagstuhl.de/opus/volltexte/2006/338 Dagstuhl Seminar Proceedings 05191 Graph Drawing http://drops.dagstuhl.de/opus/volltexte/2006/348 2 M. Jünger, S. Kobourov and P. Mutzel Matrix Zoom: A Visual Interface to Semi-external Graphs James Abello (Rutgers Univ. -
Visualizing Graphs: Optimization and Trade-Offs
Visualizing Graphs: Optimization and Trade-offs by Debajyoti Mondal A Thesis submitted to the Faculty of Graduate Studies of The University of Manitoba in partial fulfilment of the requirements of the degree of DOCTOR OF PHILOSOPHY Department of Computer Science University of Manitoba Winnipeg Copyright c 2016 by Debajyoti Mondal Thesis advisor Author Dr. Stephane Durocher Debajyoti Mondal Abstract Effective visualization of graphs is a powerful tool to help understand the rela- tionships among the graph’s underlying objects and to interact with them. Sev- eral styles for drawing graphs have emerged over the last three decades. Polyline drawing is a widely used style for drawing graphs, where each node is mapped to a distinct point in the plane and each edge is mapped to a polygonal chain between their corresponding nodes. Some common optimization criteria for such a drawing are defined in terms of area requirement, number of bends per edge, angular resolution, number of distinct line segments, edge crossings, and number of planar layers. In this thesis we develop algorithms for drawing graphs that optimize different aesthetic qualities of the drawing. Our algorithms seek to simultaneously opti- mize multiple drawing aesthetics, reveal potential trade-offs among them, and improve many previous graph drawing algorithms. We start by exploring probable trade-offs in the context of planar graphs. We prove that every n-vertex planar triangulation G with maximum degree D can be drawn with at most 2n + t 3 segments and O(8t D2t) area, where t is the − · number of leaves in a Schnyder tree of G. -
Arxiv:1712.05006V1 [Math.CO] 13 Dec 2017 Oebr2,2017
The List Linear Arboricity of Graphs Ringi Kim1 Department of Mathematical Sciences KAIST Daejeon South Korea 34141 and Luke Postle2 Department of Combinatorics and Optimization University of Waterloo Waterloo, ON Canada N2L 3G1 ABSTRACT A linear forest is a forest in which every connected component is a path. The linear arboricity of a graph G is the minimum number of linear forests of G covering all edges. In 1980, Akiyama, Exoo and Harary proposed a conjecture, known as the Linear Arboricity Conjecture (LAC), stating ∆+1 that every ∆-regular graph G has linear arboricity 2 . In 1988, Alon proved that the LAC holds asymptotically. In 1999, the list version of the LAC wa s raised by An and Wu, which is called the List Linear Arboricity Conjecture. In this article, we prove that the List Linear Arboricity Conjecture holds asymptotically. arXiv:1712.05006v1 [math.CO] 13 Dec 2017 November 23, 2017. [email protected]. [email protected]. This research was partially supported by NSERC under Discovery Grant No. 2014- 06162, the Ontario Early Researcher Awards program and the Canada Research Chairs program. 1 1 Introduction In this paper, we consider only undirected simple graphs. A linear forest is a forest in which every connected component is a path. The linear arboricity of a graph G, denoted by la(G), first introduced by Harary [13], is the minimum number of linear forests of G needed to cover all edges of G. Akiyama, Exoo and Harary [1] proposed a conjecture, known as the Linear Arboricity ∆+1 Conjecture, stating that for every ∆-regular graph G, la(G) = 2 . -
2017Sacls104
2017SACLS104 THÈSE DE DOCTORAT DE SHANDONG UNIVERSITY ET DE L’UNIVERSITÉ PARIS-SACLAY PRÉPARÉE À L’UNIVERSITÉ PARIS-SUD ÉCOLE DOCTORALE N°580 Sciences et technologies de l’information et de la communication GRADUATE SCHOOL of Mathematics Spécialité Informatique Par Mme Renyu Xu Quelques problèmes de coloration du graphe Thèse présentée et soutenue à Jinan, le 27 Mai 2017 : Composition du Jury : M. Hongjian Lai, Professeur, West Virginia University, Présidente du Jury Mme Lianying Miao, Professeure, China University of Mining and Technology, Rapporteur Mme Yinghong Ma, Professeure, Shandong Normal University, Rapporteur M. Evripidis Bampis, Professeur, Université Paris VI, Rapporteur Mme Michèle Sebag, Directrice de Recherche CNRS, Examinatrice M. Yannis Manoussakis, Professeur, Directeur de thèse M. Jianliang Wu, Professeur, CoDirecteur de thèse Contents Abstract iii Re´sume´(Abstract in French) vii Symbol Description xiii Chapter 1 Introduction 1 1.1 Basic Definitions and Notations . 3 1.2 SomeSpecialgraphs ....................... 4 1.3 Somecoloringproblemsofgraphs . 7 1.4 Mainresults............................ 21 Chapter 2 Total Coloring 25 2.1 Basic definitions and properties . 25 2.2 Planar graph G with maximum degree ∆ ≥ 8 ......... 25 2.2.1 If every 6-cycle of G contains at most one chord or chordal 6-cycles are not adjacent in G .......... 26 2.3.1 7-cycles containing at most two chords . 38 2.5 Planar graph with ∆ ≥ 7..................... 53 Chapter 3 List Coloring 65 3.1 Listvertexcoloring ........................ 65 3.2 List edge coloring and List total coloring . 68 3.2.1 Planar graph G with maximum degree ∆ ≥ 8...... 69 3.2.2 Planar graph G with maximum degree ∆ ≥ 6..... -
[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. -
Geometric Simultaneous Embeddings of a Graph and a Matching Sergio Cabello 1 Marc Van Kreveld 2 Giuseppe Liotta 3 Henk Meijer 4 Bettina Speckmann 5 Kevin Verbeek 5
Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 15, no. 1, pp. 79–96 (2011) Geometric Simultaneous Embeddings of a Graph and a Matching Sergio Cabello 1 Marc van Kreveld 2 Giuseppe Liotta 3 Henk Meijer 4 Bettina Speckmann 5 Kevin Verbeek 5 1Faculty of Mathematics and Physics, University of Ljubljana, Slovenia 2Department of Information and Computing Sciences, Utrecht University, The Netherlands 3Dipartimento di Ingegneria Elettronica e dell’Informazione, Universit`adi Perugia, Italy 4Roosevelt Academy, Middelburg, The Netherlands 5Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands Abstract The geometric simultaneous embedding problem asks whether two pla- nar graphs on the same set of vertices in the plane can be drawn using straight lines, such that each graph is plane. Geometric simultaneous em- bedding is a current topic in graph drawing and positive and negative results are known for various classes of graphs. So far only connected graphs have been considered. In this paper we present the first results for the setting where one of the graphs is a matching. In particular, we show that there exist a planar graph and a matching which do not admit a geometric simultaneous embedding. This strength- ens an analogous negative result for a planar graph and a path. On the positive side, we describe algorithms that compute a geometric simulta- neous embedding of a matching and a wheel, outerpath, or tree. Our drawing algorithms minimize the number of orientations used to draw the edges of the matching. Specifically, when embedding a matching and a tree, we can draw all matching edges horizontally. -
Relating Graph Thickness to Planar Layers and Bend Complexity∗
Relating Graph Thickness to Planar Layers and Bend Complexity∗ Stephane Durocher†1 and Debajyoti Mondal2 1 Department of Computer Science, University of Manitoba, Winnipeg, Canada [email protected] 2 Department of Computer Science, University of Manitoba, Winnipeg, Canada [email protected] Abstract The thickness of a graph G = (V, E) with n vertices is the minimum number of planar subgraphs of G whose union is G. A polyline drawing of G in R2 is a drawing Γ of G, where each vertex is mapped to a point and each edge is mapped to a polygonal chain. Bend and layer complexities are two important aesthetics of such a drawing. The bend complexity of Γ is the maximum number of bends per edge in Γ, and the layer complexity of Γ is the minimum integer r such that the set of polygonal chains in Γ can be partitioned into r disjoint sets, where each set corresponds to a planar polyline drawing. Let G be a graph of thickness t. By Fáry’s theorem, if t = 1, then G can be drawn on a single layer with bend complexity 0. A few extensions to higher thickness are known, e.g., if t = 2 (resp., t > 2), then G can be drawn on t layers with bend complexity 2 (resp., 3n + O(1)). In this paper we present an elegant extension of Fáry’s theorem to draw graphs of thickness t > 2. We first prove that thickness-t graphs can be drawn on t layers with 2.25n + O(1) bends per edge. -
On Graph Thickness, Geometric Thickness, and Separator Theorems
CCCG 2009, Vancouver, BC, August 17{19, 2009 On Graph Thickness, Geometric Thickness, and Separator Theorems Christian A. Duncan∗ Abstract fined to be the smallest number of forests whose union is G [25, 26, 27]. In other words, the minimum number We investigate the relationship between geometric of edge colors such that the subgraph induced by each thickness and the thickness, outerthickness, and ar- color is a forest of trees. In linear arboricity the result- boricity of graphs. In particular, we prove that all ing colored subgraphs must be collections of paths [2, 3]. graphs with arboricity two or outerthickness two have In outerthickness, the resulting colored subgraphs must geometric thickness O(log n). The technique used can be outerplanar graphs [16, 17]. Outerplanar graphs are be extended to other classes of graphs so long as a stan- analogous to graphs with book thickness one; however, dard separator theorem exists. For example, we can outerthickness and book thickness are not identical. apply it to show the known boundp that thickness two Another way to look at the problem is to divide the graphs have geometric thickness O( n), yielding a sim- edges of the graphs into different layers and draw the ple construction in the process. layers independently as planar graphs such that the ver- tex positions are identical in each layer. In this case, 1 Introduction the problem is to minimize the number of layers. A common related application of this problem is in VLSI In many applications of graph visualization and graph design where wires are placed in various layers to avoid theory, it is often useful to draw the edges of a graph crossings, see for example [1]. -
Thickness and Colorability of Geometric Graphs
Thickness and Colorability of Geometric Graphs Stephane Durocherx?, Ellen Gethnery, and Debajyoti Mondalx?? xDepartment of Computer Science, University of Manitoba yDepartment of Computer Science, University of Colorado Denver fdurocher,[email protected], [email protected] Abstract. The geometric thickness θ(G) of a graph G is the smallest integer t such that there exist a straight-line drawing Γ of G and a par- tition of its straight-line edges into t subsets, where each subset induces a planar drawing in Γ . Over a decade ago, Hutchinson, Shermer, and Vince proved that any n-vertex graph with geometric thickness two can have at most 6n − 18 edges, and for every n ≥ 8 they constructed a geometric thickness two graph with 6n − 20 edges. In this paper, we construct geometric thickness two graphs with 6n − 19 edges for every n ≥ 9, which improves the previously known 6n − 20 lower bound. We then construct a thickness two graph with 10 vertices that has geomet- ric thickness three, and prove that the problem of recognizing geometric thickness two graphs is NP-hard, answering two questions posed by Dil- lencourt, Eppstein and Hirschberg. Finally, we prove the NP-hardness of coloring graphs of geometric thickness t with 4t − 1 colors, which strengthens a result of McGrae and Zito, when t = 2. 1 Introduction The thickness θ(G) of a graph G is the smallest integer t such that the edges of G can be partitioned into t subsets, where each subset induces a planar graph. Since 1963, when Tutte [21] first formally introduced the notion of graph thick- ness, this property of graphs has been extensively studied for its interest from both the theoretical [2, 5, 7] and practical point of view [17, 19]. -
On the Linear K-Arboricity of Cubic Graphs
On the Linear k-arboricity of Cubic Graphs Bill Jackson Department of Mathematical Studies Goldsmiths' College London SE14 6NW, England Nicholas C. Wormald Department of Mathematics University of Melbourne Parkville, VIC 3052 Australia Abstract Bermond et al. [2] conjectured that the edge set of a cubic graph G can be partitioned into two linear k-forests, that is to say two forests whose connected components are paths of length at most k, for all k 5. We shall prove a weaker result that the statement is valid for ¸ all k 18. ¸ All graphs considered will be ¯nite. We shall refer to graphs which may contain loops or multiple edges as multigraphs and reserve the term graph for those which do not. A linear forest is a forest each of whose components are paths. The linear arboricity of a graph G, de¯ned by Harary [7], is the minimum number of linear forests required to partition E(G) and is denoted by la(G). It was shown by Akiyama, Exoo and Harary [1] that la(G) = 2 when G is cubic. A k-linear forest is a forest consisting of paths of length at most k. The k-linear arboricity of G, introduced by Bermond et al. [2], is the minimum number of k-linear forests required to partition E(G), and is denoted by lak(G). It is conjectured in [2] that if G is cubic then la5(G) 2. A partial result is obtained by Delamarre et al. [3] who show that la (G)· 2 k · 1 1 when k 2 V (G) 4. -
Drawing Graphs on Few Lines and Few Planes∗
Journal of Computational Geometry jocg.org DRAWING GRAPHS ON FEW LINES AND FEW PLANES∗ Steven Chaplick,yz Krzysztof Fleszar,§ Fabian Lipp,y{ Alexander Ravsky,k Oleg Verbitsky,∗∗ and Alexander Wolffyyy Abstract. We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight- line edges and crossing-free drawings. This problem has many relations to other challenging graph-drawing problems such as small-area or small-volume drawings, layered or track draw- ings, and drawing graphs with low visual complexity. While some facts about our problem are implicit in previous work, this is the first treatment of the problem in its full generality. Our contribution is as follows. – We show lower and upper bounds for the numbers of lines and planes needed for covering drawings of graphs in certain graph classes. In some cases our bounds are asymptotically tight; in some cases we are able to determine exact values. – We relate our parameters to standard combinatorial characteristics of graphs (such as the chromatic number, treewidth, or arboricity) and to parameters that have been studied in graph drawing (such as the track number or the number of segments ap- pearing in a drawing). – We pay special attention to planar graphs. For example, we show that there are planar graphs that can be drawn in 3-space on asymptotically fewer lines than in the plane. 1 Introduction It is well known that any graph admits a crossing-free straight-line drawing in 3-space.