DOCSLIB.ORG

Geometric Representations of Graphs with Low Polygonal Complexity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Geometric Representations of Graphs with Low Polygonal Complexity Geometric Representations of Graphs with low Polygonal Complexity vorgelegt von Diplom-Mathematiker Torsten Ueckerdt aus Berlin Von der Fakultät II – Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften – Dr. rer. nat. – genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. Jörg Liesen Berichter: Prof. Dr. Stefan Felsner Prof. Dr. Jan Kratochvil Prof. Dr. Stephen G. Kobourov Tag der wissenschaftlichen Aussprache: 4. November 2011 Berlin 2012 D 83 Preface This thesis is the fruit of my time in the Discrete Mathematics Group at TU Berlin. This group is a particularly great culture medium for mathematical plants of all kinds. Someone throws in a seed in form of a problem, a thought or an idea, and everybody can watch it either flourishing or wilting. Right from the beginning I was pleased about the possibility to work and learn within the Discrete Mathematics Group, and today I can take some nice flowers along. I want to thank my advisor Stefan Felsner for always letting me choose my prob- lems, for joining their treatment with well-trained patience and valuable ideas, and for his support of any kind. I am also thankful to Jan Kratochvil and Stephen G. Kobourov for being reviewers of my thesis. During the years, I had the opportunity to meet and work with a lot of smart and kind researchers, in particular Marie Albenque, Daniel Heldt, Andrea Hoffkamp, and Kolja Knauer, as well as, Thomas Hixon and Irina Mustata here in Berlin, Bartłomiej Bosek, Tomasz Krawczyk, Piotr Micek, and Bartosz Walczak from Jagiellonian Uni- versity in Kraków, Poland, and Stephen G. Kobourov and Muhammad Jawaherul Alam from University of Arizona. I would like to emphasize my thanks to Daniel Heldt and Kolja Knauer for our joint research for Chapter 4, but also for listening to my thoughts and discussing even the wild ones. Furthermore, I do thank Stephen G. Kobourov and Muhammad Jawaherul Alam: During their stay in Berlin we did the research for Chapter 3 in a relaxed and productive atmosphere, which I very much enjoyed. Last and not least, I very much want to thank my wife, für alle bestehenden und bestandenen Abenteuer. Thank you all! Torsten Ueckerdt Berlin, August 2011 Contents Introduction 1 1 Preliminaries 5 1.1 VertexOrderings ................................ 5 1.1.1 Degeneracy ............................... 6 1.1.2 Tree-Width............................... 7 1.1.3 Canonical Orders . 8 1.1.4 SchnyderWoods............................ 10 1.1.5 Separation-Trees and Level-i Subgraphs . 14 1.2 Orientations with Prescribed Out-Degrees . ...... 16 1.2.1 Near-Linear Time Computation . 20 1.3 Rectangle-Representations and Transversal Structures.......... 24 2 Side Contact Representations 29 2.1 Non-Rotated Representations and Schnyder Woods . ..... 35 2.1.1 Overall Complexity and Number of Segments . 49 2.2 Representations from Nesting Assignments . ..... 51 2.3 Lower Bounds on the Complexity . 58 3 Cartograms 63 3.1 Area-Universal Layouts . 67 3.2 Cartograms for Hamiltonian Maximally Planar Graphs . ...... 75 i Contents 3.2.1 One-Sided Hamiltonian Cycles . 81 3.3 Lower Bounds on the Complexity . 86 3.3.1 Hole-Free Cartograms for Planar 3-Trees . 88 3.4 Tackling 4-Connected Maximally Planar Graphs . ..... 93 4 Edge-IntersectionGraphsofGridPaths 99 4.1 The Bend-Number of Complete Bipartite Graphs . 103 4.2 The Bend-Number of Planar and Outer-Planar Graphs . 112 4.3 Fixed Degeneracy, Tree-Width, or Maximum Degree . 126 4.3.1 The Bend-Number in Terms of the Degeneracy . 127 4.3.2 The Bend-Number in Terms of the Tree-Width . 129 4.3.3 The Bend-Number in Terms of the Maximum Degree . 131 4.4 Recognizing Single-Bend Graphs is NP-Complete . 132 4.4.1 ClauseGadgets ............................ 133 4.4.2 TheReduction............................. 135 4.5 Comparison with Interval-Number and (Local) Track-Number . 138 Open Questions 143 Bibliography 147 Index 157 ii Introduction There are several ways to think of a graph and many of them involve drawing pictures. In the most classical visualization vertices are considered as points in the plane and edges as continuous curves connecting two points, such as in the top-left of Figure 1. Indeed, graph properties of eminent importance, e.g., planarity, are defined with respect to those drawings. Other popular graph visualizations include intersection representations. For exam- ple, every vertex is depicted as a point set in the plane and an edge between two vertices is described by an intersection of the corresponding point sets, such as in the bottom-left of Figure 1. In a contact representation the point set for each vertex is compact and those sets are pairwise interior disjoint. Then intersections involve only boundaries, as in the right of Figure 1, and are thus called contacts. Figure 1: A drawing, an intersection, and a contact representation of a graph. 1 Introduction Many kinds of intersection graphs have been considered, ranging back from Koebe’s “Kissing Coins Theorem” [Koe36] in 1936, up to segment representations of planar graphs due to Chapolin and Gonçalves in 2009 [CG09], and further. Within this thesis we investigate two types of intersection graphs, in both of which vertices are represented by polygonal objects in the plane. We measure the com- plexity of a polygonal object by the number of its corners. We are then particularly interested in a low polygonal complexity for every vertex, i.e., we want the maximum complexity over all vertices to be as low as possible. Chapter 2 deals with side contact representations with one simple polygon for every vertex. Every graph that admits such a representation is necessarily planar. The major part of Chapter 2 concerns hole-free rectilinear representations, i.e., those in which every side of every polygon is either horizontal or vertical, and where the union of all polygons does not leave any holes. The right of Figure 2 shows such a contact representation. We consider here maximally planar graphs only, which is a natural (and almost necessary) assumption in this setting. One of our results is a new proof that polygons of complexity 8 are always sufficient and sometimes necessary for a hole-free rectilinear representation of a maximally planar graph. In Chapter 3 we investigate what happens if we additionally prescribe the area of each and every polygon in the representation. A representation that respects a set of desired areas is known as a cartogram. For example, we prove that one can require any set of areas without increasing the worst-case maximum complexity of a hole-free rectilinear representation, i.e., 8-gons are still sufficient for every cartogram of a maximally planar graph. The second type of intersection graph is investigated in Chapter 4. In an EPG representation vertices are represented as polygonal paths with solely horizontal and vertical segments, and an edge occurs whenever two paths overlap along some part of non-zero length, i.e., neither a touching point nor a crossing causes an edge. An example of such a representation is provided in the left of Figure 2. This time, every graph admits an EPG representation. However, we again want the polygonal com- plexity, i.e., the number of corners, per path to be low. The least possible maximum complexity over all paths for a given graph is the bend-number. We give several new upper and lower bounds on the maximum bend-number for certain graph classes, such as, planar and outer-planar graphs, complete bipartite graphs, graphs of certain tree-width, maximum degree, or degeneracy. This is how the thesis is organized. Chapter 1: This chapter introduces the basic concepts and notation we use within this thesis. Section 1.1 is about vertex orderings and in particular building 2 Figure 2: A planar graph with an EPG representation on the left and a side contact representation on the right. sequences associated with them. Those building sequences underlie many of the constructive proofs presented in subsequent chapters. We define the degeneracy and the tree-width of a graph in terms of vertex orderings. For maximally planar graphs, we review the concepts of canonical orders and Schnyder woods, and outline some aspects of their close relation to each other. Furthermore, we define the separation-tree of an embedded maximally planar graph and deduce the level-i subgraphs from it. In Section 1.2 we consider orientations with prescribed out-degrees, so-called α-orientations, review their most important properties and present an algorithm that computes the minimal α-orientation in near-linear time. We close the preliminaries with Section 1.3, in which we briefly introduce rectangle-representations and transversal structures. Chapter 2: In this chapter we investigate side contact representations of planar graphs, i.e., vertices are represented by simple polygons which are pairwise in- terior disjoint, and edges correspond to side contacts. In Section 2.1 we are particularly interested in rectilinear hole-free representations with low polygo- nal complexity. We present a general method to obtain such a representation for a maximally planar graph from a Schnyder wood. In special cases, we obtain a characterization of those maximally planar graphs that admit a non-rotated rectilinear representation with complexity 4, and 6. Furthermore, we derive a new compact floor plan for maximally planar graphs. In Section 2.2 we improve a result of Sun and Sarrafzadeh [SS93] by presenting a linear-time algorithm that constructs a rectilinear representation with complexity 6 based on a nesting assignment. Our algorithm can be adjusted to construct non-rectilinear repre- sentations of complexity 5, which in particular proves their existence under the presence of a nesting assignment. At the end of this chapter in Section 2.3, we provide a general method to compute lower bounds on the complexity of side contact representations.
Load more

Recommended publications
  • On Treewidth and Graph Minors
    On Treewidth and Graph Minors Daniel John Harvey Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy February 2014 Department of Mathematics and Statistics The University of Melbourne Produced on archival quality paper ii Abstract Both treewidth and the Hadwiger number are key graph parameters in structural and al- gorithmic graph theory, especially in the theory of graph minors. For example, treewidth demarcates the two major cases of the Robertson and Seymour proof of Wagner's Con- jecture. Also, the Hadwiger number is the key measure of the structural complexity of a graph. In this thesis, we shall investigate these parameters on some interesting classes of graphs. The treewidth of a graph defines, in some sense, how \tree-like" the graph is. Treewidth is a key parameter in the algorithmic field of fixed-parameter tractability. In particular, on classes of bounded treewidth, certain NP-Hard problems can be solved in polynomial time. In structural graph theory, treewidth is of key interest due to its part in the stronger form of Robertson and Seymour's Graph Minor Structure Theorem. A key fact is that the treewidth of a graph is tied to the size of its largest grid minor. In fact, treewidth is tied to a large number of other graph structural parameters, which this thesis thoroughly investigates. In doing so, some of the tying functions between these results are improved. This thesis also determines exactly the treewidth of the line graph of a complete graph. This is a critical example in a recent paper of Marx, and improves on a recent result by Grohe and Marx.
    [Show full text]
  • Linear K-Arboricities on Trees
    Discrete Applied Mathematics 103 (2000) 281–287 View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Note Linear k-arboricities on trees Gerard J. Changa; 1, Bor-Liang Chenb, Hung-Lin Fua, Kuo-Ching Huangc; ∗;2 aDepartment of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan bDepartment of Business Administration, National Taichung Institue of Commerce, Taichung 404, Taiwan cDepartment of Applied Mathematics, Providence University, Shalu 433, Taichung, Taiwan Received 31 October 1997; revised 15 November 1999; accepted 22 November 1999 Abstract For a ÿxed positive integer k, the linear k-arboricity lak (G) of a graph G is the minimum number ‘ such that the edge set E(G) can be partitioned into ‘ disjoint sets and that each induces a subgraph whose components are paths of lengths at most k. This paper studies linear k-arboricity from an algorithmic point of view. In particular, we present a linear-time algo- rithm to determine whether a tree T has lak (T)6m. ? 2000 Elsevier Science B.V. All rights reserved. Keywords: Linear forest; Linear k-forest; Linear arboricity; Linear k-arboricity; Tree; Leaf; Penultimate vertex; Algorithm; NP-complete 1. Introduction All graphs in this paper are simple, i.e., ÿnite, undirected, loopless, and without multiple edges. A linear k-forest is a graph whose components are paths of length at most k.Alinear k-forest partition of G is a partition of the edge set E(G) into linear k-forests. The linear k-arboricity of G, denoted by lak (G), is the minimum size of a linear k-forest partition of G.
    [Show full text]
  • Minimal Acyclic Forbidden Minors for the Family of Graphs with Bounded Path-Width
    Discrete Mathematics 127 (1994) 293-304 293 North-Holland Minimal acyclic forbidden minors for the family of graphs with bounded path-width Atsushi Takahashi, Shuichi Ueno and Yoji Kajitani Department of Electrical and Electronic Engineering, Tokyo Institute of Technology. Tokyo, 152. Japan Received 27 November 1990 Revised 12 February 1992 Abstract The graphs with bounded path-width, introduced by Robertson and Seymour, and the graphs with bounded proper-path-width, introduced in this paper, are investigated. These families of graphs are minor-closed. We characterize the minimal acyclic forbidden minors for these families of graphs. We also give estimates for the numbers of minimal forbidden minors and for the numbers of vertices of the largest minimal forbidden minors for these families of graphs. 1. Introduction Graphs we consider are finite and undirected, but may have loops and multiple edges unless otherwise specified. A graph H is a minor of a graph G if H is isomorphic to a graph obtained from a subgraph of G by contracting edges. A family 9 of graphs is said to be minor-closed if the following condition holds: If GEB and H is a minor of G then H EF. A graph G is a minimal forbidden minor for a minor-closed family F of graphs if G#8 and any proper minor of G is in 9. Robertson and Seymour proved the following deep theorems. Theorem 1.1 (Robertson and Seymour [15]). Every minor-closed family of graphs has a finite number of minimal forbidden minors. Theorem 1.2 (Robertson and Seymour [14]).
    [Show full text]
  • Orienting Fully Dynamic Graphs with Worst-Case Time Bounds⋆
    Orienting Fully Dynamic Graphs with Worst-Case Time Bounds? Tsvi Kopelowitz1??, Robert Krauthgamer2 ???, Ely Porat3, and Shay Solomon4y 1 University of Michigan, [email protected] 2 Weizmann Institute of Science, [email protected] 3 Bar-Ilan University, [email protected] 4 Weizmann Institute of Science, [email protected] Abstract. In edge orientations, the goal is usually to orient (direct) the edges of an undirected network (modeled by a graph) such that all out- degrees are bounded. When the network is fully dynamic, i.e., admits edge insertions and deletions, we wish to maintain such an orientation while keeping a tab on the update time. Low out-degree orientations turned out to be a surprisingly useful tool for managing networks. Brodal and Fagerberg (1999) initiated the study of the edge orientation problem in terms of the graph's arboricity, which is very natural in this context. Their solution achieves a constant out-degree and a logarithmic amortized update time for all graphs with constant arboricity, which include all planar and excluded-minor graphs. It remained an open ques- tion { first proposed by Brodal and Fagerberg, later by Erickson and others { to obtain similar bounds with worst-case update time. We address this 15 year old question by providing a simple algorithm with worst-case bounds that nearly match the previous amortized bounds. Our algorithm is based on a new approach of maintaining a combina- torial invariant, and achieves a logarithmic out-degree with logarithmic worst-case update times. This result has applications to various dynamic network problems such as maintaining a maximal matching, where we obtain logarithmic worst-case update time compared to a similar amor- tized update time of Neiman and Solomon (2013).
    [Show full text]
  • Faster Sublinear Approximation of the Number of K-Cliques in Low-Arboricity Graphs
    Faster sublinear approximation of the number of k-cliques in low-arboricity graphs Talya Eden ✯ Dana Ron ❸ C. Seshadhri ❹ Abstract science [10, 49, 40, 58, 7], with a wide variety of applica- Given query access to an undirected graph G, we consider tions [37, 13, 53, 18, 44, 8, 6, 31, 54, 38, 27, 57, 30, 39]. the problem of computing a (1 ε)-approximation of the This problem has seen a resurgence of interest because number of k-cliques in G. The± standard query model for general graphs allows for degree queries, neighbor queries, of its importance in analyzing massive real-world graphs and pair queries. Let n be the number of vertices, m be (like social networks and biological networks). There the number of edges, and nk be the number of k-cliques. are a number of clever algorithms for exactly counting Previous work by Eden, Ron and Seshadhri (STOC 2018) ∗ n mk/2 k-cliques using matrix multiplications [49, 26] or combi- gives an O ( 1 + )-time algorithm for this problem n /k nk k natorial methods [58]. However, the complexity of these (we use O∗( ) to suppress poly(log n, 1/ε,kk) dependencies). algorithms grows with mΘ(k), where m is the number of · Moreover, this bound is nearly optimal when the expression edges in the graph. is sublinear in the size of the graph. Our motivation is to circumvent this lower bound, by A line of recent work has considered this question parameterizing the complexity in terms of graph arboricity. from a sublinear approximation perspective [20, 24].
    [Show full text]
  • Characterizations of Graphs Without Certain Small Minors by J. Zachary
    Characterizations of Graphs Without Certain Small Minors By J. Zachary Gaslowitz Dissertation Submitted to the Faculty of the Graduate School of Vanderbilt University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Mathematics May 11, 2018 Nashville, Tennessee Approved: Mark Ellingham, Ph.D. Paul Edelman, Ph.D. Bruce Hughes, Ph.D. Mike Mihalik, Ph.D. Jerry Spinrad, Ph.D. Dong Ye, Ph.D. TABLE OF CONTENTS Page 1 Introduction . 2 2 Previous Work . 5 2.1 Planar Graphs . 5 2.2 Robertson and Seymour's Graph Minor Project . 7 2.2.1 Well-Quasi-Orderings . 7 2.2.2 Tree Decomposition and Treewidth . 8 2.2.3 Grids and Other Graphs with Large Treewidth . 10 2.2.4 The Structure Theorem and Graph Minor Theorem . 11 2.3 Graphs Without K2;t as a Minor . 15 2.3.1 Outerplanar and K2;3-Minor-Free Graphs . 15 2.3.2 Edge-Density for K2;t-Minor-Free Graphs . 16 2.3.3 On the Structure of K2;t-Minor-Free Graphs . 17 3 Algorithmic Aspects of Graph Minor Theory . 21 3.1 Theoretical Results . 21 3.2 Practical Graph Minor Containment . 22 4 Characterization and Enumeration of 4-Connected K2;5-Minor-Free Graphs 25 4.1 Preliminary Definitions . 25 4.2 Characterization . 30 4.3 Enumeration . 38 5 Characterization of Planar 4-Connected DW6-minor-free Graphs . 51 6 Future Directions . 91 BIBLIOGRAPHY . 93 1 Chapter 1 Introduction All graphs in this paper are finite and simple. Given a graph G, the vertex set of G is denoted V (G) and the edge set is denoted E(G).
    [Show full text]
  • 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.
    [Show full text]
  • Sphere-Cut Decompositions and Dominating Sets in Planar Graphs
    Sphere-cut Decompositions and Dominating Sets in Planar Graphs Michalis Samaris R.N. 201314 Scientific committee: Dimitrios M. Thilikos, Professor, Dep. of Mathematics, National and Kapodistrian University of Athens. Supervisor: Stavros G. Kolliopoulos, Dimitrios M. Thilikos, Associate Professor, Professor, Dep. of Informatics and Dep. of Mathematics, National and Telecommunications, National and Kapodistrian University of Athens. Kapodistrian University of Athens. white Lefteris M. Kirousis, Professor, Dep. of Mathematics, National and Kapodistrian University of Athens. Aposunjèseic sfairik¸n tom¸n kai σύνοla kuriarqÐac se epÐpeda γραφήματa Miχάλης Σάμαρης A.M. 201314 Τριμελής Epiτροπή: Δημήτρioc M. Jhlυκός, Epiblèpwn: Kajhγητής, Tm. Majhmatik¸n, E.K.P.A. Δημήτρioc M. Jhlυκός, Staύρoc G. Kolliόποuloc, Kajhγητής tou Τμήμatoc Anaπληρωτής Kajhγητής, Tm. Plhroforiκής Majhmatik¸n tou PanepisthmÐou kai Thl/ni¸n, E.K.P.A. Ajhn¸n Leutèrhc M. Kuroύσης, white Kajhγητής, Tm. Majhmatik¸n, E.K.P.A. PerÐlhyh 'Ena σημαντικό apotèlesma sth JewrÐa Γραφημάτwn apoteleÐ h apόdeixh thc eikasÐac tou Wagner από touc Neil Robertson kai Paul D. Seymour. sth σειρά ergasi¸n ‘Ελλάσσοna Γραφήματα’ apo to 1983 e¸c to 2011. H eikasÐa αυτή lèei όti sthn κλάση twn γραφημάtwn den υπάρχει άπειρη antialusÐda ¸c proc th sqèsh twn ελλασόnwn γραφημάτwn. H JewrÐa pou αναπτύχθηκε gia thn απόδειξη αυτής thc eikasÐac eÐqe kai èqei ακόμα σημαντικό antÐktupo tόσο sthn δομική όσο kai sthn algoriθμική JewrÐa Γραφημάτwn, άλλα kai se άλλα pedÐa όπως h Παραμετρική Poλυπλοκόthta. Sta πλάιsia thc απόδειξης oi suggrafeÐc eiσήγαγαν kai nèec paramètrouc πλά- touc. Se autèc ήτan h κλαδοαποσύνθεση kai to κλαδοπλάτoc ενός γραφήματoc. H παράμετρος αυτή χρησιμοποιήθηκε idiaÐtera sto σχεδιασμό algorÐjmwn kai sthn χρήση thc τεχνικής ‘διαίρει kai basÐleue’.
    [Show full text]
  • Linearity of Grid Minors in Treewidth with Applications Through Bidimensionality∗
    Linearity of Grid Minors in Treewidth with Applications through Bidimensionality∗ Erik D. Demaine† MohammadTaghi Hajiaghayi† Abstract We prove that any H-minor-free graph, for a fixed graph H, of treewidth w has an Ω(w) × Ω(w) grid graph as a minor. Thus grid minors suffice to certify that H-minor-free graphs have large treewidth, up to constant factors. This strong relationship was previously known for the special cases of planar graphs and bounded-genus graphs, and is known not to hold for general graphs. The approach of this paper can be viewed more generally as a framework for extending combinatorial results on planar graphs to hold on H-minor-free graphs for any fixed H. Our result has many combinatorial consequences on bidimensionality theory, parameter-treewidth bounds, separator theorems, and bounded local treewidth; each of these combinatorial results has several algorithmic consequences including subexponential fixed-parameter algorithms and approximation algorithms. 1 Introduction The r × r grid graph1 is the canonical planar graph of treewidth Θ(r). In particular, an important result of Robertson, Seymour, and Thomas [38] is that every planar graph of treewidth w has an Ω(w) × Ω(w) grid graph as a minor. Thus every planar graph of large treewidth has a grid minor certifying that its treewidth is almost as large (up to constant factors). In their Graph Minor Theory, Robertson and Seymour [36] have generalized this result in some sense to any graph excluding a fixed minor: for every graph H and integer r > 0, there is an integer w > 0 such that every H-minor-free graph with treewidth at least w has an r × r grid graph as a minor.
    [Show full text]
  • On Star and Caterpillar Arboricity
    Discrete Mathematics 309 (2009) 3694–3702 www.elsevier.com/locate/disc On star and caterpillar arboricity Daniel Gonc¸alvesa,∗, Pascal Ochemb a LIRMM UMR 5506, CNRS, Universite´ Montpelier 2, 161 rue Ada, 34392 Montpellier Cedex 5, France b LRI UMR 8623, CNRS, Universite´ Paris-Sud, Batˆ 490, 91405 Orsay Cedex, France Received 31 October 2005; accepted 18 January 2008 Available online 10 March 2008 Abstract We give new bounds on the star arboricity and the caterpillar arboricity of planar graphs with given girth. One of them answers an open problem of Gyarf´ as´ and West: there exist planar graphs with track number 4. We also provide new NP-complete problems. c 2008 Elsevier B.V. All rights reserved. Keywords: NP-completeness; Partitioning problems; Edge coloring 1. Introduction Many graph parameters in the literature are defined as the minimum size of a partition of the edges of the graph such that each part induces a graph of a given class C. The most common is the chromatic index χ 0.G/, in this case C is the class of graphs with maximum degree one. Vizing [18] proved that χ 0.G/ either equals ∆.G/ or ∆.G/ C 1, where ∆.G/ denotes the maximum degree of G. Deciding whether χ 0.G/ D 3 is shown to be NP-complete for general graphs in [13]. The arboricity a.G/ is another well studied parameter, for which C is the class of forests. In [15], Nash-Williams proved that: jE.H/j a.G/ D max (1) H⊆G jV .H/j − 1 with the maximum being over all the subgraphs H D .E.H/; V .H// of G.
    [Show full text]
  • Tree-Decomposition Graph Minor Theory and Algorithmic Implications
    DIPLOMARBEIT Tree-Decomposition Graph Minor Theory and Algorithmic Implications Ausgeführt am Institut für Diskrete Mathematik und Geometrie der Technischen Universität Wien unter Anleitung von Univ.Prof. Dipl.-Ing. Dr.techn. Michael Drmota durch Miriam Heinz, B.Sc. Matrikelnummer: 0625661 Baumgartenstraße 53 1140 Wien Datum Unterschrift Preface The focus of this thesis is the concept of tree-decomposition. A tree-decomposition of a graph G is a representation of G in a tree-like structure. From this structure it is possible to deduce certain connectivity properties of G. Such information can be used to construct efficient algorithms to solve problems on G. Sometimes problems which are NP-hard in general are solvable in polynomial or even linear time when restricted to trees. Employing the tree-like structure of tree-decompositions these algorithms for trees can be adapted to graphs of bounded tree-width. This results in many important algorithmic applications of tree-decomposition. The concept of tree-decomposition also proves to be useful in the study of fundamental questions in graph theory. It was used extensively by Robertson and Seymour in their seminal work on Wagner’s conjecture. Their Graph Minors series of papers spans more than 500 pages and results in a proof of the graph minor theorem, settling Wagner’s conjecture in 2004. However, it is not only the proof of this deep and powerful theorem which merits mention. Also the concepts and tools developed for the proof have had a major impact on the field of graph theory. Tree-decomposition is one of these spin-offs. Therefore, we will study both its use in the context of graph minor theory and its several algorithmic implications.
    [Show full text]
  • 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.
    [Show full text]