Grundy Distinguishes Treewidth from Pathwidth
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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. -
Treewidth I (Algorithms and Networks)
Treewidth Algorithms and Networks Overview • Historic introduction: Series parallel graphs • Dynamic programming on trees • Dynamic programming on series parallel graphs • Treewidth • Dynamic programming on graphs of small treewidth • Finding tree decompositions 2 Treewidth Computing the Resistance With the Laws of Ohm = + 1 1 1 1789-1854 R R1 R2 = + R R1 R2 R1 R2 R1 Two resistors in series R2 Two resistors in parallel 3 Treewidth Repeated use of the rules 6 6 5 2 2 1 7 Has resistance 4 1/6 + 1/2 = 1/(1.5) 1.5 + 1.5 + 5 = 8 1 + 7 = 8 1/8 + 1/8 = 1/4 4 Treewidth 5 6 6 5 2 2 1 7 A tree structure tree A 5 6 P S 2 P 6 P 2 7 S Treewidth 1 Carry on! • Internal structure of 6 6 5 graph can be forgotten 2 2 once we know essential information 1 7 about it! 4 ¼ + ¼ = ½ 6 Treewidth Using tree structures for solving hard problems on graphs 1 • Network is ‘series parallel graph’ • 196*, 197*: many problems that are hard for general graphs are easy for e.g.: – Trees NP-complete – Series parallel graphs • Many well-known problems Linear / polynomial time computable 7 Treewidth Weighted Independent Set • Independent set: set of vertices that are pair wise non-adjacent. • Weighted independent set – Given: Graph G=(V,E), weight w(v) for each vertex v. – Question: What is the maximum total weight of an independent set in G? • NP-complete 8 Treewidth Weighted Independent Set on Trees • On trees, this problem can be solved in linear time with dynamic programming. -
Randomly Coloring Graphs of Logarithmically Bounded Pathwidth Shai Vardi
Randomly coloring graphs of logarithmically bounded pathwidth Shai Vardi To cite this version: Shai Vardi. Randomly coloring graphs of logarithmically bounded pathwidth. 2018. hal-01832102 HAL Id: hal-01832102 https://hal.archives-ouvertes.fr/hal-01832102 Preprint submitted on 6 Jul 2018 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. Randomly coloring graphs of logarithmically bounded pathwidth Shai Vardi∗ Abstract We consider the problem of sampling a proper k-coloring of a graph of maximal degree ∆ uniformly at random. We describe a new Markov chain for sampling colorings, and show that it mixes rapidly on graphs of logarithmically bounded pathwidth if k ≥ (1 + )∆, for any > 0, using a new hybrid paths argument. ∗California Institute of Technology, Pasadena, CA, 91125, USA. E-mail: [email protected]. 1 Introduction A (proper) k-coloring of a graph G = (V; E) is an assignment σ : V ! f1; : : : ; kg such that neighboring vertices have different colors. We consider the problem of sampling (almost) uniformly at random from the space of all k-colorings of a graph.1 The problem has received considerable attention from the computer science community in recent years, e.g., [12, 17, 24, 26, 33, 44, 45]. -
Interval Edge-Colorings of Graphs
University of Central Florida STARS Electronic Theses and Dissertations, 2004-2019 2016 Interval Edge-Colorings of Graphs Austin Foster University of Central Florida Part of the Mathematics Commons Find similar works at: https://stars.library.ucf.edu/etd University of Central Florida Libraries http://library.ucf.edu This Masters Thesis (Open Access) is brought to you for free and open access by STARS. It has been accepted for inclusion in Electronic Theses and Dissertations, 2004-2019 by an authorized administrator of STARS. For more information, please contact [email protected]. STARS Citation Foster, Austin, "Interval Edge-Colorings of Graphs" (2016). Electronic Theses and Dissertations, 2004-2019. 5133. https://stars.library.ucf.edu/etd/5133 INTERVAL EDGE-COLORINGS OF GRAPHS by AUSTIN JAMES FOSTER B.S. University of Central Florida, 2015 A thesis submitted in partial fulfilment of the requirements for the degree of Master of Science in the Department of Mathematics in the College of Sciences at the University of Central Florida Orlando, Florida Summer Term 2016 Major Professor: Zixia Song ABSTRACT A proper edge-coloring of a graph G by positive integers is called an interval edge-coloring if the colors assigned to the edges incident to any vertex in G are consecutive (i.e., those colors form an interval of integers). The notion of interval edge-colorings was first introduced by Asratian and Kamalian in 1987, motivated by the problem of finding compact school timetables. In 1992, Hansen described another scenario using interval edge-colorings to schedule parent-teacher con- ferences so that every person’s conferences occur in consecutive slots. -
David Gries, 2018 Graph Coloring A
Graph coloring A coloring of an undirected graph is an assignment of a color to each node so that adja- cent nodes have different colors. The graph to the right, taken from Wikipedia, is known as the Petersen graph, after Julius Petersen, who discussed some of its properties in 1898. It has been colored with 3 colors. It can’t be colored with one or two. The Petersen graph has both K5 and bipartite graph K3,3, so it is not planar. That’s all you have to know about the Petersen graph. But if you are at all interested in what mathemati- cians and computer scientists do, visit the Wikipedia page for Petersen graph. This discussion on graph coloring is important not so much for what it says about the four-color theorem but what it says about proofs by computers, for the proof of the four-color theorem was just about the first one to use a computer and sparked a lot of controversy. Kempe’s flawed proof that four colors suffice to color a planar graph Thoughts about graph coloring appear to have sprung up in England around 1850 when people attempted to color maps, which can be represented by planar graphs in which the nodes are countries and adjacent countries have a directed edge between them. Francis Guthrie conjectured that four colors would suffice. In 1879, Alfred Kemp, a barrister in London, published a proof in the American Journal of Mathematics that only four colors were needed to color a planar graph. Eleven years later, P.J. -
Effective and Efficient Dynamic Graph Coloring
Effective and Efficient Dynamic Graph Coloring Long Yuanx, Lu Qinz, Xuemin Linx, Lijun Changy, and Wenjie Zhangx x The University of New South Wales, Australia zCentre for Quantum Computation & Intelligent Systems, University of Technology, Sydney, Australia y The University of Sydney, Australia x{longyuan,lxue,zhangw}@cse.unsw.edu.au; [email protected]; [email protected] ABSTRACT (1) Nucleic Acid Sequence Design in Biochemical Networks. Given Graph coloring is a fundamental graph problem that is widely ap- a set of nucleic acids, a dependency graph is a graph in which each plied in a variety of applications. The aim of graph coloring is to vertex is a nucleotide and two vertices are connected if the two minimize the number of colors used to color the vertices in a graph nucleotides form a base pair in at least one of the nucleic acids. such that no two incident vertices have the same color. Existing The problem of finding a nucleic acid sequence that is compatible solutions for graph coloring mainly focus on computing a good col- with the set of nucleic acids can be modelled as a graph coloring oring for a static graph. However, since many real-world graphs are problem on a dependency graph [57]. highly dynamic, in this paper, we aim to incrementally maintain the (2) Air Traffic Flow Management. In air traffic flow management, graph coloring when the graph is dynamically updated. We target the air traffic flow can be considered as a graph in which each vertex on two goals: high effectiveness and high efficiency. -
Coloring Problems in Graph Theory Kacy Messerschmidt Iowa State University
Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2018 Coloring problems in graph theory Kacy Messerschmidt Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Mathematics Commons Recommended Citation Messerschmidt, Kacy, "Coloring problems in graph theory" (2018). Graduate Theses and Dissertations. 16639. https://lib.dr.iastate.edu/etd/16639 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Coloring problems in graph theory by Kacy Messerschmidt A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Bernard Lidick´y,Major Professor Steve Butler Ryan Martin James Rossmanith Michael Young The student author, whose presentation of the scholarship herein was approved by the program of study committee, is solely responsible for the content of this dissertation. The Graduate College will ensure this dissertation is globally accessible and will not permit alterations after a degree is conferred. Iowa State University Ames, Iowa 2018 Copyright c Kacy Messerschmidt, 2018. All rights reserved. TABLE OF CONTENTS LIST OF FIGURES iv ACKNOWLEDGEMENTS vi ABSTRACT vii 1. INTRODUCTION1 2. DEFINITIONS3 2.1 Basics . .3 2.2 Graph theory . .3 2.3 Graph coloring . .5 2.3.1 Packing coloring . .6 2.3.2 Improper coloring . -
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]). -
Arxiv:2006.06067V2 [Math.CO] 4 Jul 2021
Treewidth versus clique number. I. Graph classes with a forbidden structure∗† Cl´ement Dallard1 Martin Milaniˇc1 Kenny Storgelˇ 2 1 FAMNIT and IAM, University of Primorska, Koper, Slovenia 2 Faculty of Information Studies, Novo mesto, Slovenia [email protected] [email protected] [email protected] Treewidth is an important graph invariant, relevant for both structural and algo- rithmic reasons. A necessary condition for a graph class to have bounded treewidth is the absence of large cliques. We study graph classes closed under taking induced subgraphs in which this condition is also sufficient, which we call (tw,ω)-bounded. Such graph classes are known to have useful algorithmic applications related to variants of the clique and k-coloring problems. We consider six well-known graph containment relations: the minor, topological minor, subgraph, induced minor, in- duced topological minor, and induced subgraph relations. For each of them, we give a complete characterization of the graphs H for which the class of graphs excluding H is (tw,ω)-bounded. Our results yield an infinite family of χ-bounded induced-minor-closed graph classes and imply that the class of 1-perfectly orientable graphs is (tw,ω)-bounded, leading to linear-time algorithms for k-coloring 1-perfectly orientable graphs for every fixed k. This answers a question of Breˇsar, Hartinger, Kos, and Milaniˇc from 2018, and one of Beisegel, Chudnovsky, Gurvich, Milaniˇc, and Servatius from 2019, respectively. We also reveal some further algorithmic implications of (tw,ω)- boundedness related to list k-coloring and clique problems. In addition, we propose a question about the complexity of the Maximum Weight Independent Set prob- lem in (tw,ω)-bounded graph classes and prove that the problem is polynomial-time solvable in every class of graphs excluding a fixed star as an induced minor. -
Structural Parameterizations of Clique Coloring
Structural Parameterizations of Clique Coloring Lars Jaffke University of Bergen, Norway lars.jaff[email protected] Paloma T. Lima University of Bergen, Norway [email protected] Geevarghese Philip Chennai Mathematical Institute, India UMI ReLaX, Chennai, India [email protected] Abstract A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with q colors. For fixed q ≥ 2, we give an O?(qtw)-time algorithm when the input graph is given together with one of its tree decompositions of width tw. We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is XP parameterized by clique-width. 2012 ACM Subject Classification Mathematics of computing → Graph coloring Keywords and phrases clique coloring, treewidth, clique-width, structural parameterization, Strong Exponential Time Hypothesis Digital Object Identifier 10.4230/LIPIcs.MFCS.2020.49 Related Version A full version of this paper is available at https://arxiv.org/abs/2005.04733. Funding Lars Jaffke: Supported by the Trond Mohn Foundation (TMS). Acknowledgements The work was partially done while L. J. and P. T. L. were visiting Chennai Mathematical Institute. 1 Introduction Vertex coloring problems are central in algorithmic graph theory, and appear in many variants. One of these is Clique Coloring, which given a graph G and an integer k asks whether G has a clique coloring with k colors, i.e. -
Clique-Width and Tree-Width of Sparse Graphs
Clique-width and tree-width of sparse graphs Bruno Courcelle Labri, CNRS and Bordeaux University∗ 33405 Talence, France email: [email protected] June 10, 2015 Abstract Tree-width and clique-width are two important graph complexity mea- sures that serve as parameters in many fixed-parameter tractable (FPT) algorithms. The same classes of sparse graphs, in particular of planar graphs and of graphs of bounded degree have bounded tree-width and bounded clique-width. We prove that, for sparse graphs, clique-width is polynomially bounded in terms of tree-width. For planar and incidence graphs, we establish linear bounds. Our motivation is the construction of FPT algorithms based on automata that take as input the algebraic terms denoting graphs of bounded clique-width. These algorithms can check properties of graphs of bounded tree-width expressed by monadic second- order formulas written with edge set quantifications. We give an algorithm that transforms tree-decompositions into clique-width terms that match the proved upper-bounds. keywords: tree-width; clique-width; graph de- composition; sparse graph; incidence graph Introduction Tree-width and clique-width are important graph complexity measures that oc- cur as parameters in many fixed-parameter tractable (FPT) algorithms [7, 9, 11, 12, 14]. They are also important for the theory of graph structuring and for the description of graph classes by forbidden subgraphs, minors and vertex-minors. Both notions are based on certain hierarchical graph decompositions, and the associated FPT algorithms use dynamic programming on these decompositions. To be usable, they need input graphs of "small" tree-width or clique-width, that furthermore are given with the relevant decompositions. -
Handout on Vertex Separators and Low Tree-Width K-Partition
Handout on vertex separators and low tree-width k-partition January 12 and 19, 2012 Given a graph G(V; E) and a set of vertices S ⊂ V , an S-flap is the set of vertices in a connected component of the graph induced on V n S. A set S is a vertex separator if no S-flap has more than n=p 2 vertices. Lipton and Tarjan showed that every planar graph has a separator of size O( n). This was generalized by Alon, Seymour and Thomas to any family of graphs that excludes some fixed (arbitrary) subgraph H as a minor. Theorem 1 There a polynomial time algorithm that given a parameter h and an n vertex graphp G(V; E) either outputs a Kh minor, or outputs a vertex separator of size at most h hn. Corollary 2 Let G(V; E) be an arbitrary graph with nop Kh minor, and let W ⊂ V . Then one can find in polynomial time a set S of at most h hn vertices such that every S-flap contains at most jW j=2 vertices from W . Proof: The proof given in class for Theorem 1 easily extends to this setting. 2 p Corollary 3 Every graph with no Kh as a minor has treewidth O(h hn). Moreover, a tree decomposition with this treewidth can be found in polynomial time. Proof: We have seen an algorithm that given a graph of treewidth p constructs a tree decomposition of treewidth 8p. Usingp Corollary 2, that algorithm can be modified to give a tree decomposition of treewidth 8h hn in our case, and do so in polynomial time.