Finer Tight Bounds for Coloring on Clique-Width
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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. -
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. -
Packing and Covering Dense Graphs
Packing and Covering Dense Graphs Noga Alon ∗ Yair Caro y Raphael Yuster z Abstract Let d be a positive integer. A graph G is called d-divisible if d divides the degree of each vertex of G. G is called nowhere d-divisible if no degree of a vertex of G is divisible by d. For a graph H, gcd(H) denotes the greatest common divisor of the degrees of the vertices of H. The H-packing number of G is the maximum number of pairwise edge disjoint copies of H in G. The H-covering number of G is the minimum number of copies of H in G whose union covers all edges of G. Our main result is the following: For every fixed graph H with gcd(H) = d, there exist positive constants (H) and N(H) such that if G is a graph with at least N(H) vertices and has minimum degree at least (1 − (H)) G , then the H-packing number of G and the H-covering number of G can be computed j j in polynomial time. Furthermore, if G is either d-divisible or nowhere d-divisible, then there is a closed formula for the H-packing number of G, and the H-covering number of G. Further extensions and solutions to related problems are also given. 1 Introduction All graphs considered here are finite, undirected and simple, unless otherwise noted. For the standard graph-theoretic terminology the reader is referred to [1]. Let H be a graph without isolated vertices. An H-covering of a graph G is a set L = G1;:::;Gs of subgraphs of G, where f g each subgraph is isomorphic to H, and every edge of G appears in at least one member of L. -
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. -
Embedding Trees in Dense Graphs
Embedding trees in dense graphs Václav Rozhoň February 14, 2019 joint work with T. Klimo¹ová and D. Piguet Václav Rozhoň Embedding trees in dense graphs February 14, 2019 1 / 11 Alternative definition: substructures in graphs Extremal graph theory Definition (Extremal graph theory, Bollobás 1976) Extremal graph theory, in its strictest sense, is a branch of graph theory developed and loved by Hungarians. Václav Rozhoň Embedding trees in dense graphs February 14, 2019 2 / 11 Extremal graph theory Definition (Extremal graph theory, Bollobás 1976) Extremal graph theory, in its strictest sense, is a branch of graph theory developed and loved by Hungarians. Alternative definition: substructures in graphs Václav Rozhoň Embedding trees in dense graphs February 14, 2019 2 / 11 Density of edges vs. density of triangles (Razborov 2008) (image from the book of Lovász) Extremal graph theory Theorem (Mantel 1907) Graph G has n vertices. If G has more than n2=4 edges then it contains a triangle. Generalisations? Václav Rozhoň Embedding trees in dense graphs February 14, 2019 3 / 11 Extremal graph theory Theorem (Mantel 1907) Graph G has n vertices. If G has more than n2=4 edges then it contains a triangle. Generalisations? Density of edges vs. density of triangles (Razborov 2008) (image from the book of Lovász) Václav Rozhoň Embedding trees in dense graphs February 14, 2019 3 / 11 3=2 The answer for C4 is of order n , lower bound via finite projective planes. What is the answer for trees? Extremal graph theory Theorem (Mantel 1907) Graph G has n vertices. If G has more than n2=4 edges then it contains a triangle. -
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 . -
Fractional Arboricity, Strength, and Principal Partitions in Graphs and Matroids
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Applied Mathematics 40 (1992) 285-302 285 North-Holland Fractional arboricity, strength, and principal partitions in graphs and matroids Paul A. Catlin Wayne State University, Detroit, MI 48202. USA Jerrold W. Grossman Oakland University, Rochester, MI 48309, USA Arthur M. Hobbs* Texas A & M University, College Station, TX 77843, USA Hong-Jian Lai** West Virginia University, Morgantown, WV 26506, USA Received 15 February 1989 Revised 15 November 1989 Abstract Catlin, P.A., J.W. Grossman, A.M. Hobbs and H.-J. Lai, Fractional arboricity, strength, and principal partitions in graphs and matroids, Discrete Applied Mathematics 40 (1992) 285-302. In a 1983 paper, D. Gusfield introduced a function which is called (following W.H. Cunningham, 1985) the strength of a graph or matroid. In terms of a graph G with edge set E(G) and at least one link, this is the function v(G) = mmFr9c) IFl/(w(G F) - w(G)), where the minimum is taken over all subsets F of E(G) such that w(G F), the number of components of G - F, is at least w(G) + 1. In a 1986 paper, C. Payan introduced thefractional arboricity of a graph or matroid. In terms of a graph G with edge set E(G) and at least one link this function is y(G) = maxHrC lE(H)I/(l V(H)1 -w(H)), where H runs over Correspondence to: Professor A.M. Hobbs, Department of Mathematics, Texas A & M University, College Station, TX 77843, USA. -
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. -
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.