Lecture 10: April 20, 2005 Perfect Graphs
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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. -
I?'!'-Comparability Graphs
Discrete Mathematics 74 (1989) 173-200 173 North-Holland I?‘!‘-COMPARABILITYGRAPHS C.T. HOANG Department of Computer Science, Rutgers University, New Brunswick, NJ 08903, U.S.A. and B.A. REED Discrete Mathematics Group, Bell Communications Research, Morristown, NJ 07950, U.S.A. In 1981, Chv&tal defined the class of perfectly orderable graphs. This class of perfect graphs contains the comparability graphs. In this paper, we introduce a new class of perfectly orderable graphs, the &comparability graphs. This class generalizes comparability graphs in a natural way. We also prove a decomposition theorem which leads to a structural characteriza- tion of &comparability graphs. Using this characterization, we develop a polynomial-time recognition algorithm and polynomial-time algorithms for the clique and colouring problems for &comparability graphs. 1. Introduction A natural way to color the vertices of a graph is to put them in some order v*cv~<” l < v, and then assign colors in the following manner: (1) Consider the vertxes sequentially (in the sequence given by the order) (2) Assign to each Vi the smallest color (positive integer) not used on any neighbor vi of Vi with j c i. We shall call this the greedy coloring algorithm. An order < of a graph G is perfect if for each subgraph H of G, the greedy algorithm applied to H, gives an optimal coloring of H. An obstruction in a ordered graph (G, <) is a set of four vertices {a, b, c, d} with edges ab, bc, cd (and no other edge) and a c b, d CC. Clearly, if < is a perfect order on G, then (G, C) contains no obstruction (because the chromatic number of an obstruction is twu, but the greedy coloring algorithm uses three colors). -
Arxiv:2106.16130V1 [Math.CO] 30 Jun 2021 in the Special Case of Cyclohedra, and by Cardinal, Langerman and P´Erez-Lantero [5] in the Special Case of Tree Associahedra
LAGOS 2021 Bounds on the Diameter of Graph Associahedra Jean Cardinal1;4 Universit´elibre de Bruxelles (ULB) Lionel Pournin2;4 Mario Valencia-Pabon3;4 LIPN, Universit´eSorbonne Paris Nord Abstract Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph G encodes the combinatorics of search trees on G, defined recursively by a root r together with search trees on each of the connected components of G − r. In particular, the skeleton of the graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. It is known that the diameter of the associahedra of paths of length n, the classical associahedra, is 2n − 6 for a large enough n. We give a tight bound of Θ(m) on the diameter of trivially perfect graph associahedra on m edges. We consider the maximum diameter of associahedra of graphs on n vertices and of given tree-depth, treewidth, or pathwidth, and give lower and upper bounds as a function of these parameters. Finally, we prove that the maximum diameter of associahedra of graphs of pathwidth two is Θ(n log n). Keywords: generalized permutohedra, graph associahedra, tree-depth, treewidth, pathwidth 1 Introduction The vertices and edges of a polyhedron form a graph whose diameter (often referred to as the diameter of the polyhedron for short) is related to a number of computational problems. For instance, the question of how large the diameter of a polyhedron arises naturally from the study of linear programming and the simplex algorithm (see, for instance [27] and references therein). -
The Strong Perfect Graph Theorem
Annals of Mathematics, 164 (2006), 51–229 The strong perfect graph theorem ∗ ∗ By Maria Chudnovsky, Neil Robertson, Paul Seymour, * ∗∗∗ and Robin Thomas Abstract A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. The “strong perfect graph conjecture” (Berge, 1961) asserts that a graph is perfect if and only if it is Berge. A stronger conjecture was made recently by Conforti, Cornu´ejols and Vuˇskovi´c — that every Berge graph either falls into one of a few basic classes, or admits one of a few kinds of separation (designed so that a minimum counterexample to Berge’s conjecture cannot have either of these properties). In this paper we prove both of these conjectures. 1. Introduction We begin with definitions of some of our terms which may be nonstandard. All graphs in this paper are finite and simple. The complement G of a graph G has the same vertex set as G, and distinct vertices u, v are adjacent in G just when they are not adjacent in G.Ahole of G is an induced subgraph of G which is a cycle of length at least 4. An antihole of G is an induced subgraph of G whose complement is a hole in G. A graph G is Berge if every hole and antihole of G has even length. A clique in G is a subset X of V (G) such that every two members of X are adjacent. -
Box Representations of Embedded Graphs
Box representations of embedded graphs Louis Esperet CNRS, Laboratoire G-SCOP, Grenoble, France S´eminairede G´eom´etrieAlgorithmique et Combinatoire, Paris March 2017 Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. -
Improved Ramsey-Type Results in Comparability Graphs
Improved Ramsey-type results in comparability graphs D´anielKor´andi ∗ Istv´anTomon ∗ Abstract Several discrete geometry problems are equivalent to estimating the size of the largest homogeneous sets in graphs that happen to be the union of few comparability graphs. An important observation for such results is that if G is an n-vertex graph that is the union of r comparability (or more generally, perfect) graphs, then either G or its complement contains a clique of size n1=(r+1). This bound is known to be tight for r = 1. The question whether it is optimal for r ≥ 2 was studied by Dumitrescu and T´oth.We prove that it is essentially best possible for r = 2, as well: we introduce a probabilistic construction of two comparability graphs on n vertices, whose union contains no clique or independent set of size n1=3+o(1). Using similar ideas, we can also construct a graph G that is the union of r comparability graphs, and neither G, nor its complement contains a complete bipartite graph with parts of size cn (log n)r . With this, we improve a result of Fox and Pach. 1 Introduction An old problem of Larman, Matouˇsek,Pach and T¨or}ocsik[11, 14] asks for the largest m = m(n) such that among any n convex sets in the plane, there are m that are pairwise disjoint or m that pairwise intersect. Considering the disjointness graph G, whose vertices correspond to the convex sets and edges correspond to disjoint pairs, this problem asks the Ramsey-type question of estimating the largest clique or independent set in G. -
Results on Independent Sets in Categorical Products of Graphs, The
Results on independent sets in categorical products of graphs, the ultimate categorical independence ratio and the ultimate categorical independent domination ratio Wing-Kai Hon1, Ton Kloks, Hsiang-Hsuan Liu1, Sheung-Hung Poon1, and Yue-Li Wang2 1 Department of Computer Science National Tsing Hua University, Taiwan {wkhon,hhliu,spoon}@cs.nthu.edu.tw 2 Department of Information Management National Taiwan University of Science and Technology [email protected] Abstract. We show that there are polynomial-time algorithms to compute maximum independent sets in the categorical products of two cographs and two splitgraphs. The ultimate categorical independence ratio of a k k graph G is defined as limk→ α(G )/n . The ultimate categorical inde- pendence ratio is polynomial for cographs, permutation graphs, interval graphs, graphs of bounded treewidth∞ and splitgraphs. When G is a planar graph of maximal degree three then α(G K4) is NP-complete. We present a PTAS for the ultimate categorical independence× ratio of planar graphs. We present an O∗(nn/3) exact, exponential algorithm for general graphs. We prove that the ultimate categorical independent domination ratio for complete multipartite graphs is zero, except when the graph is complete bipartite with color classes of equal size (in which case it is1/2). 1 Introduction Let G and H be two graphs. The categorical product also travels under the guise of tensor product, or direct product, or Kronecker product, and even more names arXiv:1306.1656v1 [cs.DM] 7 Jun 2013 have been given to it. It is defined as follows. It is a graph, denoted as G H. -
P 4-Colorings and P 4-Bipartite Graphs Chinh T
P_4-Colorings and P_4-Bipartite Graphs Chinh T. Hoàng, van Bang Le To cite this version: Chinh T. Hoàng, van Bang Le. P_4-Colorings and P_4-Bipartite Graphs. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2001, 4 (2), pp.109-122. hal-00958951 HAL Id: hal-00958951 https://hal.inria.fr/hal-00958951 Submitted on 13 Mar 2014 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. Discrete Mathematics and Theoretical Computer Science 4, 2001, 109–122 P4-Free Colorings and P4-Bipartite Graphs Ch´ınh T. Hoang` 1† and Van Bang Le2‡ 1Department of Physics and Computing, Wilfrid Laurier University, 75 University Ave. W., Waterloo, Ontario N2L 3C5, Canada 2Fachbereich Informatik, Universitat¨ Rostock, Albert-Einstein-Straße 21, D-18051 Rostock, Germany received May 19, 1999, revised November 25, 2000, accepted December 15, 2000. A vertex partition of a graph into disjoint subsets Vis is said to be a P4-free coloring if each color class Vi induces a subgraph without a chordless path on four vertices (denoted by P4). Examples of P4-free 2-colorable graphs (also called P4-bipartite graphs) include parity graphs and graphs with “few” P4s like P4-reducible and P4-sparse graphs. -
On Sources in Comparability Graphs, with Applications Stephan Olariu Old Dominion University
Old Dominion University ODU Digital Commons Computer Science Faculty Publications Computer Science 1992 On Sources in Comparability Graphs, With Applications Stephan Olariu Old Dominion University Follow this and additional works at: https://digitalcommons.odu.edu/computerscience_fac_pubs Part of the Applied Mathematics Commons, and the Computer Sciences Commons Repository Citation Olariu, Stephan, "On Sources in Comparability Graphs, With Applications" (1992). Computer Science Faculty Publications. 128. https://digitalcommons.odu.edu/computerscience_fac_pubs/128 Original Publication Citation Olariu, S. (1992). On sources in comparability graphs, with applications. Discrete Mathematics, 110(1-3), 289-292. doi:10.1016/ 0012-365x(92)90721-q This Article is brought to you for free and open access by the Computer Science at ODU Digital Commons. It has been accepted for inclusion in Computer Science Faculty Publications by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. Discrete Mathematics 110 (1992) 2X9-292 289 North-Holland Note On sources in comparability graphs, with applications S. Olariu Department of Computer Science, Old Dominion University, Norfolk, VA 235294162, USA Received 31 October 1989 Abstract Olariu, S., On sources in comparability graphs, with applications, Discrete Mathematics 110 (1992) 289-292. We characterize sources in comparability graphs and show that our result provides a unifying look at two recent results about interval graphs. An orientation 0 of a graph G is obtained by assigning unique directions to its edges. To simplify notation, we write xy E 0, whenever the edge xy receives the direction from x to y. A vertex w is called a source whenever VW E 0 for no vertex II in G. -
Research Topics in Graph Theory and Its Applications
Research Topics in Graph Theory and Its Applications Research Topics in Graph Theory and Its Applications By Vadim Zverovich Research Topics in Graph Theory and Its Applications By Vadim Zverovich This book first published 2019 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2019 by Vadim Zverovich All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-5275-3533-9 ISBN (13): 978-1-5275-3533-6 Disclaimer: Any statements in this book might be fictitious and they represent the author's opinion. In memory of my son, Vladik (1987 { 2000) . Contents Preface xi 1 α-Discrepancy and Strong Perfectness 1 1.1 Background and Aims . 1 1.2 α-Discrepancy . 4 1.3 Strongly Perfect Graphs . 7 1.4 Computational Aspects . 10 1.5 Beneficiaries and Dissemination . 13 1.6 References . 14 1.7 Resources and Work Plan . 16 1.8 Reviewers' Reports . 19 1.9 PI's Response to Reviewers' Comments . 30 1.10 Summary of Key Lessons . 32 2 Braess' Paradox in Transport Networks 35 2.1 Background and Aims . 35 2.2 Braess' Paradox and Methodology . 40 2.3 Computational Aspects of Braess' Paradox . 44 2.4 Braess-like Situations in Rail Networks . 48 2.5 References . 49 2.6 Beneficiaries, Resources and Work Plan . -
Combinatorial Optimization and Recognition of Graph Classes with Applications to Related Models
Combinatorial Optimization and Recognition of Graph Classes with Applications to Related Models Von der Fakult¨at fur¨ Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Diplom-Mathematiker George B. Mertzios aus Thessaloniki, Griechenland Berichter: Privat Dozent Dr. Walter Unger (Betreuer) Professor Dr. Berthold V¨ocking (Zweitbetreuer) Professor Dr. Dieter Rautenbach Tag der mundlichen¨ Prufung:¨ Montag, den 30. November 2009 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfugbar.¨ Abstract This thesis mainly deals with the structure of some classes of perfect graphs that have been widely investigated, due to both their interesting structure and their numerous applications. By exploiting the structure of these graph classes, we provide solutions to some open problems on them (in both the affirmative and negative), along with some new representation models that enable the design of new efficient algorithms. In particular, we first investigate the classes of interval and proper interval graphs, and especially, path problems on them. These classes of graphs have been extensively studied and they find many applications in several fields and disciplines such as genetics, molecular biology, scheduling, VLSI design, archaeology, and psychology, among others. Although the Hamiltonian path problem is well known to be linearly solvable on interval graphs, the complexity status of the longest path problem, which is the most natural optimization version of the Hamiltonian path problem, was an open question. We present the first polynomial algorithm for this problem with running time O(n4). Furthermore, we introduce a matrix representation for both interval and proper interval graphs, called the Normal Interval Representation (NIR) and the Stair Normal Interval Representation (SNIR) matrix, respectively. -
A Constructive Formalization of the Weak Perfect Graph Theorem
A Constructive Formalization of the Weak Perfect Graph Theorem Abhishek Kr Singh1 and Raja Natarajan2 1 Tata Institute of Fundamental Research, Mumbai, India [email protected] 2 Tata Institute of Fundamental Research, Mumbai, India [email protected] Abstract The Perfect Graph Theorems are important results in graph theory describing the relationship between clique number ω(G) and chromatic number χ(G) of a graph G. A graph G is called perfect if χ(H) = ω(H) for every induced subgraph H of G. The Strong Perfect Graph Theorem (SPGT) states that a graph is perfect if and only if it does not contain an odd hole (or an odd anti-hole) as its induced subgraph. The Weak Perfect Graph Theorem (WPGT) states that a graph is perfect if and only if its complement is perfect. In this paper, we present a formal framework for working with finite simple graphs. We model finite simple graphs in the Coq Proof Assistant by representing its vertices as a finite set over a countably infinite domain. We argue that this approach provides a formal framework in which it is convenient to work with different types of graph constructions (or expansions) involved in the proof of the Lovász Replication Lemma (LRL), which is also the key result used in the proof of Weak Perfect Graph Theorem. Finally, we use this setting to develop a constructive formalization of the Weak Perfect Graph Theorem. Digital Object Identifier 10.4230/LIPIcs.CPP.2020. 1 Introduction The chromatic number χ(G) of a graph G is the minimum number of colours needed to colour the vertices of G so that no two adjacent vertices get the same colour.