Orthogonal Tree Decompositions of Graphs
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On Grounded L-Graphs and Their Relatives Arxiv:1808.04148V1
On grounded L-graphs and their relatives Vít Jelínek∗ Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Prague, Czechia [email protected] Martin Töpfery Institute of Science and Technology, Klosterneuburg, Austria [email protected] August 14, 2018 Abstract We consider the graph classes Grounded-L and Grounded-fL; L g corresponding to graphs that admit an intersection representation by L-shaped curves (or L-shaped and L -shaped curves, respectively), where additionally the topmost points of each curve are assumed to belong to a common horizontal line. We prove that Grounded-L graphs admit an equivalent characterisation in terms of vertex ordering with forbidden patterns. We also compare these classes to related intersection classes, such as the grounded segment graphs, the monotone L-graphs (a.k.a. max point-tolerance graphs), or the outer-1-string graphs. We give con- structions showing that these classes are all distinct and satisfy only arXiv:1808.04148v1 [math.CO] 13 Aug 2018 trivial or previously known inclusions. ∗Supported by the project 16-01602Y of the Czech Science Foundation and by project Neuron Impuls of the Neuron Fund for Support of Science. yThis project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 665385. 1 1 Introduction An intersection representation of a graph G = (V; E) is a map that assigns to every vertex x 2 V a set sx in such a way that two vertices x and y are adjacent if and only if the two corresponding sets sx and sy intersect. -
Separators in Region Intersection Graphs
Separators in region intersection graphs James R. Lee∗ Abstract For undirected graphs G V; E and G0 V0; E0 , say that G is a region intersection ¹ º ¹ º graph over G0 if there is a family of connected subsets Ru V0 : u V of G0 such that f ⊆ 2 g u; v E Ru Rv , . f g 2 () \ ; We show if G0 excludes the complete graph Kh as a minor for some h > 1, then every region intersection graph G over G0 with m edges has a balanced separator with at most chpm nodes, where ch is a constant depending only on h. If G additionally has uniformly bounded vertex degrees, then such a separator is found by spectral partitioning. A string graph is the intersection graph of continuous arcs in the plane. String graphs are precisely region intersection graphs over planar graphs. Thus the preceding result implies that every string graph with m edges has a balanced separator of size O pm . This bound is optimal, as it generalizes the planar separator theorem. It confirms a conjecture¹ º of Fox and Pach (2010), and improves over the O pm log m bound of Matoušek (2013). ¹ º Contents 1 Introduction 2 1.1 Balanced separators and extremal spread.........................3 1.2 Eigenvalues and L2-extremal spread............................7 1.3 Additional applications....................................7 1.4 Preliminaries..........................................7 2 Vertex separators and conformal graph metrics8 2.1 Conformal graphs.......................................8 2.2 Padded partitions and random separators......................... 10 3 Multi-flows, congestion, and crossings 12 3.1 Duality between conformal metrics and multi-flows.................. -
Treewidth-Erickson.Pdf
Computational Topology (Jeff Erickson) Treewidth Or il y avait des graines terribles sur la planète du petit prince . c’étaient les graines de baobabs. Le sol de la planète en était infesté. Or un baobab, si l’on s’y prend trop tard, on ne peut jamais plus s’en débarrasser. Il encombre toute la planète. Il la perfore de ses racines. Et si la planète est trop petite, et si les baobabs sont trop nombreux, ils la font éclater. [Now there were some terrible seeds on the planet that was the home of the little prince; and these were the seeds of the baobab. The soil of that planet was infested with them. A baobab is something you will never, never be able to get rid of if you attend to it too late. It spreads over the entire planet. It bores clear through it with its roots. And if the planet is too small, and the baobabs are too many, they split it in pieces.] — Antoine de Saint-Exupéry (translated by Katherine Woods) Le Petit Prince [The Little Prince] (1943) 11 Treewidth In this lecture, I will introduce a graph parameter called treewidth, that generalizes the property of having small separators. Intuitively, a graph has small treewidth if it can be recursively decomposed into small subgraphs that have small overlap, or even more intuitively, if the graph resembles a ‘fat tree’. Many problems that are NP-hard for general graphs can be solved in polynomial time for graphs with small treewidth. Graphs embedded on surfaces of small genus do not necessarily have small treewidth, but they can be covered by a small number of subgraphs, each with small treewidth. -
A Complete Anytime Algorithm for Treewidth
UAI 2004 GOGATE & DECHTER 201 A Complete Anytime Algorithm for Treewidth Vibhav Gogate and Rina Dechter School of Information and Computer Science, University of California, Irvine, CA 92967 fvgogate,[email protected] Abstract many algorithms that solve NP-hard problems in Arti¯cial Intelligence, Operations Research, In this paper, we present a Branch and Bound Circuit Design etc. are exponential only in the algorithm called QuickBB for computing the treewidth. Examples of such algorithm are Bucket treewidth of an undirected graph. This al- elimination [Dechter, 1999] and junction-tree elimina- gorithm performs a search in the space of tion [Lauritzen and Spiegelhalter, 1988] for Bayesian perfect elimination ordering of vertices of Networks and Constraint Networks. These algo- the graph. The algorithm uses novel prun- rithms operate in two steps: (1) constructing a ing and propagation techniques which are good tree-decomposition and (2) solving the prob- derived from the theory of graph minors lem on this tree-decomposition, with the second and graph isomorphism. We present a new step generally exponential in the treewidth of the algorithm called minor-min-width for com- tree-decomposition computed in step 1. In this puting a lower bound on treewidth that is paper, we present a complete anytime algorithm used within the branch and bound algo- called QuickBB for computing a tree-decomposition rithm and which improves over earlier avail- of a graph that can feed into any algorithm like able lower bounds. Empirical evaluation of Bucket elimination [Dechter, 1999] that requires a QuickBB on randomly generated graphs and tree-decomposition. benchmarks in Graph Coloring and Bayesian The majority of recent literature on Networks shows that it is consistently bet- the treewidth problem is devoted to ¯nd- ter than complete algorithms like Quick- ing constant factor approximation algo- Tree [Shoikhet and Geiger, 1997] in terms of rithms [Amir, 2001, Becker and Geiger, 2001]. -
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. -
A Polynomial Excluded-Minor Approximation of Treedepth
A Polynomial Excluded-Minor Approximation of Treedepth Ken-ichi Kawarabayashi∗ Benjamin Rossmany National Institute of Informatics University of Toronto k [email protected] [email protected] November 1, 2017 Abstract Treedepth is a well-studied graph invariant in the family of \width measures" that includes treewidth and pathwidth. Understanding these invariants in terms of excluded minors has been an active area of research. The recent Grid Minor Theorem of Chekuri and Chuzhoy [12] establishes that treewidth is polynomially approximated by the largest k × k grid minor. In this paper, we give a similar polynomial excluded-minor approximation for treedepth in terms of three basic obstructions: grids, tree, and paths. Specifically, we show that there is a constant c such that every graph of treedepth ≥ kc contains one of the following minors (each of treedepth ≥ k): • the k × k grid, • the complete binary tree of height k, • the path of order 2k. Let us point out that we cannot drop any of the above graphs for our purpose. Moreover, given a graph G we can, in randomized polynomial time, find either an embedding of one of these minors or conclude that treedepth of G is at most kc. This result has potential applications in a variety of settings where bounded treedepth plays a role. In addition to some graph structural applications, we describe a surprising application in circuit complexity and finite model theory from recent work of the second author [28]. 1 Introduction Treedepth is a well-studied graph invariant with several equivalent definitions. It appears in the literature under various names including vertex ranking number [30], ordered chromatic number [17], and minimum elimination tree height [23], before being systematically studied under the name treedepth by Ossona de Mendes and Neˇsetˇril[20]. -
Geometric Intersection Patterns and the Theory of Topological Graphs
Geometric Intersection Patterns and the Theory of Topological Graphs J´anosPach∗ Abstract. The intersection graph of a set system S is a graph on the vertex set S, in which two vertices are connected by an edge if and only if the corresponding sets have nonempty intersection. It was shown by Tietze (1905) that every finite graph is the intersection graph of 3-dimensional convex polytopes. The analogous statement is false in any fixed dimension if the polytopes are allowed to have only a bounded number of faces or are replaced by simple geometric objects that can be described in terms of a bounded number of real parameters. Intersection graphs of various classes of geometric objects, even in the plane, have interesting structural and extremal properties. We survey problems and results on geometric intersection graphs and, more gener- ally, intersection patterns. Many of the questions discussed were originally raised by Berge, Erd}os,Gr¨unbaum, Hadwiger, Tur´an,and others in the context of classical topol- ogy, graph theory, and combinatorics (related, e.g., to Helly's theorem, Ramsey theory, perfect graphs). The rapid development of computational geometry and graph drawing algorithms in the last couple of decades gave further impetus to research in this field. A topological graph is a graph drawn in the plane so that its vertices are represented by points and its edges by possibly intersecting simple continuous curves connecting the corresponding point pairs. We give applications of the results concerning intersection patterns in the theory of topological graphs. Mathematics Subject Classification (2010). Primary 05C35; Secondary 05C62, 52C10. -
Layered Separators in Minor-Closed Graph Classes with Applications
Layered Separators in Minor-Closed Graph Classes with Appli- cations Vida Dujmovi´c y Pat Morinz David R. Wood x Abstract. Graph separators are a ubiquitous tool in graph theory and computer science. However, in some applications, their usefulness is limited by the fact that the separator can be as large as Ω(pn) in graphs with n vertices. This is the case for planar graphs, and more generally, for proper minor-closed classes. We study a special type of graph separator, called a layered separator, which may have linear size in n, but has bounded size with respect to a different measure, called the width. We prove, for example, that planar graphs and graphs of bounded Euler genus admit layered separators of bounded width. More generally, we characterise the minor-closed classes that admit layered separators of bounded width as those that exclude a fixed apex graph as a minor. We use layered separators to prove (log n) bounds for a number of problems where (pn) O O was a long-standing previous best bound. This includes the nonrepetitive chromatic number and queue-number of graphs with bounded Euler genus. We extend these results with a (log n) bound on the nonrepetitive chromatic number of graphs excluding a fixed topological O minor, and a logO(1) n bound on the queue-number of graphs excluding a fixed minor. Only for planar graphs were logO(1) n bounds previously known. Our results imply that every n-vertex graph excluding a fixed minor has a 3-dimensional grid drawing with n logO(1) n volume, whereas the previous best bound was (n3=2). -
From Planar Graphs to Higher Dimension Lucas Isenmann
From planar graphs to higher dimension Lucas Isenmann To cite this version: Lucas Isenmann. From planar graphs to higher dimension. Discrete Mathematics [cs.DM]. Université Montpellier, 2019. English. NNT : 2019MONTS142. tel-02931761 HAL Id: tel-02931761 https://tel.archives-ouvertes.fr/tel-02931761 Submitted on 7 Sep 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE POUR OBTENIR LE GRADE DE DOCTEUR DE L’UNIVERSITÉ DE MONTPELLIER En informatique École doctorale I2S Unité de recherche LIRMM UMR 5506 Des graphes planaires vers des dimensions supérieures Présentée par Lucas Isenmann Décembre 2019 Sous la direction de Daniel GONÇALVES Devant le jury composé de Jean CARDINAL, Professeur, Université Libre de Bruxelles Rapporteur Nicolas BONICHON, Maitre de conférences, Université de Bordeaux Examinateur William PUECH, Professeur des universités, Université de de Montpellier Examinateur Nicolas TROTIGNON, Directeur de recherche, ENS de Lyon Examinateur Contents 1 Dushnik-Miller dimension of some geometric complexes 17 1.1 Introduction . 17 1.1.1 Basic notions . 17 1.1.2 Delaunay graphs and their variants . 19 1.1.3 Dushnik-Miller dimension and representations . 22 1.1.4 State of the art . 26 1.1.5 Organization of the chapter . -
On the Chromatic Number of Disjointness Graphs of Curves
On the Chromatic Number of Disjointness Graphs of Curves János Pach École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland Rényi Institute, Budapest, Hungary janos.pach@epfl.ch István Tomon École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland istvan.tomon@epfl.ch Abstract Let ω(G) and χ(G) denote the clique number and chromatic number of a graph G, respectively. The disjointness graph of a family of curves (continuous arcs in the plane) is the graph whose vertices correspond to the curves and in which two vertices are joined by an edge if and only if the corresponding curves are disjoint. A curve is called x-monotone if every vertical line intersects it in at most one point. An x-monotone curve is grounded if its left endpoint lies on the y-axis. We prove that if G is the disjointness graph of a family of grounded x-monotone curves such that k+1 ω(G) = k, then χ(G) ≤ 2 . If we only require that every curve is x-monotone and intersects the k+1 k+2 y-axis, then we have χ(G) ≤ 2 3 . Both of these bounds are best possible. The construction showing the tightness of the last result settles a 25 years old problem: it yields that there exist Kk-free disjointness graphs of x-monotone curves such that any proper coloring of them uses at least Ω(k4) colors. This matches the upper bound up to a constant factor. 2012 ACM Subject Classification Mathematics of computing → Graph theory Keywords and phrases string graph, chromatic number, intersection graph Digital Object Identifier 10.4230/LIPIcs.SoCG.2019.54 Related Version https://arxiv.org/abs/1811.09158 Funding János Pach: Research partially supported by Swiss National Science Foundation grants no. -
Distributed Minimum Vertex Coloring and Maximum Independent Set in Chordal Graphs∗
Distributed Minimum Vertex Coloring and Maximum Independent Set in Chordal Graphs∗ Christian Konrady Viktor Zamaraevz Abstract We give deterministic distributed (1 + )-approximation algorithms for Minimum Vertex Color- ing and Maximum Independent Set on chordal graphs in the LOCAL model. Our coloring algorithm 1 1 1 ∗ runs in O( log n) rounds, and our independent set algorithm has a runtime of O( log( ) log n) 1 rounds. For coloring, existing lower bounds imply that the dependencies on and log n are best 1 possible. For independent set, we prove that Ω( ) rounds are necessary. Both our algorithms make use of a tree decomposition of the input chordal graph. They iteratively peel off interval subgraphs, which are identified via the tree decomposition of the input graph, thereby partitioning the vertex set into O(log n) layers. For coloring, each interval graph is colored independently, which results in various coloring conflicts between the layers. These conflicts are then resolved in a separate phase, using the particular structure of our partitioning. 1 For independent set, only the first O(log ) layers are required as they already contain a large enough independent set. We develop a (1 + )-approximation maximum independent set algorithm for interval graphs, which we then apply to those layers. This work raises the question as to how useful tree decompositions are for distributed comput- ing. 1 Introduction The LOCAL Model. In the LOCAL model of distributed computation, the input graph G = (V; E) represents a communication network, where every network node hosts a computational entity. Nodes have unique IDs. A distributed algorithm is executed on all network nodes simultaneously and proceeds in discrete rounds. -
Approximate Tree Decompositions of Planar Graphs in Linear Time
Approximate Tree Decompositions of Planar Graphs in Linear Time Frank Kammer∗ and Torsten Tholey∗ Institut f¨ur Informatik, Universit¨at Augsburg, Germany Abstract Many algorithms have been developed for NP-hard problems on graphs with small treewidth k. For example, all problems that are expressible in linear extended monadic second order can be solved in linear time on graphs of bounded treewidth. It turns out that the bottleneck of many algorithms for NP-hard problems is the computation of a tree decomposi- tion of width O(k). In particular, by the bidimensional theory, there are many linear extended monadic second order problems that can be solved on n-vertex planar graphs with treewidth k in a time linear in n and subexponential in k if a tree decomposition of width O(k) can be found in such a time. We present the first algorithm that, on n-vertex planar graphs with treewidth k, finds a tree decomposition of width O(k) in such a time. In more detail, our algorithm has a running time of O(nk2 log k). We show the result as a special case of a result concerning so-called weighted treewidth of weighted graphs. Keywords: (weighted) treewidth, branchwidth, rank-width, planar graph, linear time, bidimensionality. ACM classification: F.2.2; G.2.2 1 Introduction The treewidth, extensively studied by Robertson and Seymour [22], is one of the basic parameters in graph theory. Intuitively, the treewidth measures the similarity of a graph to a tree by means of a so-called tree decomposition. A tree decomposition of width r—defined precisely in the beginning of Section 2— arXiv:1104.2275v5 [cs.DS] 14 May 2016 is a decomposition of a graph G into small subgraphs part of a so-called bag such that each bag contains at most r + 1 vertices and such that the bags are connected by a tree-like structure.