DOCSLIB.ORG

Complexity of Generalized Colourings of Chordal Graphs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Complexity of Generalized Colourings of Chordal Graphs COMPLEXITY OF GENERALIZED COLOURINGS OF CHORDAL GRAPHS by Juraj Stacho M.Sc., Comenius University, Slovakia, 2005 a thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the School of Computing Science c Juraj Stacho 2008 SIMON FRASER UNIVERSITY Spring 2008 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. APPROVAL Name: Juraj Stacho Degree: Doctor of Philosophy Title of thesis: Complexity of Generalized Colourings of Chordal Graphs Examining Committee: Dr. Joseph G. Peters Chair Dr. Pavol Hell, Senior Supervisor Dr. Ramesh Krishnamurti, Supervisor Dr. G´abor Tardos, SFU Examiner Dr. Ross M. McConnell, External Examiner, Computer Science Department, Colorado State University Date Approved: ii Abstract The generalized graph colouring problem (GCOL) for a fixed integer k, and fixed classes of graphs ,..., (usually describing some common graph properties), is to decide, for P1 Pk a given graph G, whether the vertex set of G can be partitioned into sets V1,...,Vk such that, for each i, the induced subgraph of G on V belongs to . It can be seen that GCOL i Pi generalizes many natural colouring and partitioning problems on graphs. In this thesis, we focus on generalized colouring problems in chordal graphs. The struc- ture of chordal graphs is known to allow solving many difficult combinatorial problems, such as the graph colouring, maximum clique and others, in polynomial, and in many cases in linear time. Our study of generalized colouring problems focuses on those problems in which the sets are characterized by a single forbidden induced subgraph. We show, that Pi for k = 2, all such problems where the forbidden graphs have at most three vertices are polynomial time solvable in chordal graphs, whereas, it is known that almost all of them are NP -complete in general. On the other hand, we show infinite families of such problems which are NP -complete in chordal graphs. By combining a polynomial algorithm and an NP -completeness proof, we answer a question of Broersma, Fomin, Neˇsetˇril and Woeginger about the complexity of the so-called subcolouring problem on chordal graphs. Additionally, we explain, how some of these results generalize to particular subclasses of chordal graphs, and we show a complete forbidden subgraph characterization for the so-called monopolar partitions of chordal graphs. Finally, in the last part of the thesis, we focus on a different type of colouring problem – injective colouring. We describe several algorithmic and (in-)approximability results for injective colourings in the class of chordal graphs and its subclasses. In the process, we correct a result of Agnarsson et al. on inapproximability of the chromatic number of the square of a split graph. Keywords: chordal graphs; graph colouring; subcolouring; injective colouring; forbidden subgraph characterization; polynomial time algorithms Subject Terms: Graph Theory; Graph Coloring; Graph Algorithms; Perfect Graphs; Homomorphisms (Mathematics); Graph Grammars iii Mojim rodiˇcom Milanovi a Darine To my parents Milan and Darina iv Acknowledgments The author would like to seize the opportunity to express his deepest gratitude to his thesis advisor Dr. Pavol Hell for his guidance, enthusiasm, his profound insight, and broad knowledge, and most importantly for his perpetual support during author’s graduate studies at Simon Fraser University. The author also wishes to thank his parents for their unconditional support throughout his entire academic studies and wishes to dedicate this thesis to them. v Contents Approval ii Abstract iii Dedication iv Acknowledgments v Contents vi List of Figures viii List of Tables ix List of Algorithms ix 1 Introduction 1 1.1 Definitions...................................... 4 1.2 Chordalgraphs ................................... 7 1.3 Relatedclasses .................................. 11 1.4 Polynomialtimecases ............................. 15 2 Tools 22 2.1 MatrixPartitions................................ 22 2.2 Dynamicprogramming.............................. 24 2.3 Greedyalgorithms ................................ 32 2.4 GraphGrammars.................................. 34 2.5 Treewidth and Monadic Second Order Logic . ...... 38 3 Polar colourings of Chordal Graphs 40 3.1 MonopolarChordalGraphs . 40 3.2 PolarChordalGraphs .............................. 53 vi 4 Forbidden Subgraphs for Monopolarity 63 4.1 Thetwo-connectedcase .. .. .. .. .. .. .. .. 64 4.2 Thegeneralcase .................................. 68 5 Subcolourings of Chordal Graphs 89 5.1 2-subcolourings ................................. 90 5.1.1 Subcolouringdigraph . 90 5.1.2 Algorithm.................................. 96 5.2 3-subcolourings ................................. 101 5.3 k-subcolourings...................................105 6 Pk-transversals of Chordal Graphs 108 6.1 Stable P4-transversals ...............................108 6.2 Forcinggraphs ................................... 112 6.3 Stable P5-transversals ...............................117 6.4 Pj-free Pk-transversals ...............................120 6.5 Kj-free Pk-transversals...............................124 7 Other cases 126 7.1 StronglyChordalGraphs . 135 7.2 ChordalComparabilityGraphs . 138 8 Injective Colourings 143 8.1 Basicproperties................................. 144 8.2 Hardnessandapproximationresults . 146 8.3 Exactalgorithmicresults . 151 8.3.1 Injectivestructure . .. .. .. .. .. .. .. 153 8.3.2 Computing χi(G)inchordalgraphs . .154 8.3.3 Bridgelesschordalgraphs . 156 8.3.4 Perfectly tree-dominated graphs . 157 9 Conclusions 160 Bibliography 162 vii List of Figures 1.1 All minimal precoloured forbidden induced subgraphs of split graphs. 18 2.1 The stubborn problem, its complement, and two NP -complete cases. 23 2.2 An example of a hyperedge replacement grammar generating all partial 3-trees. 36 3.1 Rules1-8. ..................................... 44 3.2 An illustration of the monopolarity recognition algorithm. ........... 45 4.1 All minimal list non-extendable forbidden induced subgraphs for monopolar- ityof2-connectedchordalgraphs.. 65 4.2 TherulesofthegrammarΓ.. 69 4.3 Additional rules of the grammar Γ′. ....................... 69 4.4 Example of a derivation of the grammar. ...... 70 4.5 Hypergraphs (part 1) for the proof of Theorem 4.9. ........ 83 4.6 Hypergraphs (part 2) for the proof of Theorem 4.9. ........ 84 4.7 Hypergraphs (part 3) for the proof of Theorem 4.9. ........ 86 5.1 Illustrating the case when there is an arc u, v in (G). ........... 91 h i Dc 5.2 The graph H and sample 3-subcolourings of H. ................103 5.3 The graph (X)withasample3-subcolouring. .103 G 5.4 The graphs Fk for small k, and the construction from Theorem 5.15. 106 6.1 The graphs Gi and Yj................................109 6.2 The cotree for the graph G S . ........................111 i − i 6.3 Forcing graphs for the Pk-transversalproblem. .112 k 6.4 Forcing graphs Hj fortheproofofTheorem6.1. .115 k k 6.5 Forcing graphs Bj and Dj for the proof of Theorem 6.1. 116 ′ ′ 6.6 The graphs Yj and Gi. ..............................117 ′ ′′ ′′′ 6.7 The graphs G (G ) and G for the Pj-free Pk-transversal problem. 120 ∗ 6.8 The graph G for the Kj-free Pk-transversal problem. 124 7.1 Matrices for selected GCOL problems in chordal graphs. ..........126 7.2 Transforming a transitive orientation into a uniform transitive orientation. 140 viii List of Tables 1.1 A summary of complexities of selected graph problems. .......... 14 9.1 A summary of complexity results in chordal graphs and general graphs. 160 List of Algorithms 1.1 Lexicographic Breadth-First search. ......... 10 3.1 Monopolar graph recognition. ...... 43 3.2 Constructingreducedlists. ...... 43 3.3 TheRules1-8andtheBlockRule. 44 3.4 Extracting a monopolar partition. ....... 46 3.5 Testingwhetheracliqueisnice. ...... 61 3.6 Polargraphrecognition.. ..... 62 5.1 The test for 2-subcolourability of G......................... 99 5.2 The test whether Pu,v is (z, w)-colourable.. .100 5.3 The test whether T is ( )colourable... .. .. .. .. .. ..100 x − 5.4 The test whether Tx is(+)colourable.. .101 7.1 Cardinality Lexicographic Breadth-First search. .............139 ix Chapter 1 Introduction The graph colouring problem is probably one of the oldest problems of graph theory. Here, the task is to label the vertices of a given graph using as few colours as possible in such a way that no adjacent vertices receive identical labels. The origins of this problem can be traced all the way back to the famous problem of four colours, formulated by Francis Guthrie in 1852, which asks whether any map of countries can be coloured using at most four colours in such a way that no two countries sharing a boundary use the same colour. As it turned out later, this is just a special case of the graph colouring problem in planar graphs. Since then, graph colouring and its variants have appeared in many different guises in combinatorics, optimization, discrete geometry, set theory, and many other branches of discrete mathematics. Also, many practical problems such as time tabling, sequencing, and scheduling are related to the problem of graph colouring. Unfortunately, graph colouring is one of the first known NP -complete problems. This tells us that it is unlikely that an efficient algorithm for graph colouring exists, where, of course, by efficient, we mean running in polynomial time (possibly also randomized). This, however, does not mean that the problem
Load more

Recommended publications
  • Lecture 10: April 20, 2005 Perfect Graphs
    Re-revised notes 4-22-2005 10pm CMSC 27400-1/37200-1 Combinatorics and Probability Spring 2005 Lecture 10: April 20, 2005 Instructor: L´aszl´oBabai Scribe: Raghav Kulkarni TA SCHEDULE: TA sessions are held in Ryerson-255, Monday, Tuesday and Thursday 5:30{6:30pm. INSTRUCTOR'S EMAIL: [email protected] TA's EMAIL: [email protected], [email protected] IMPORTANT: Take-home test Friday, April 29, due Monday, May 2, before class. Perfect Graphs k 1=k Shannon capacity of a graph G is: Θ(G) := limk (α(G )) : !1 Exercise 10.1 Show that α(G) χ(G): (G is the complement of G:) ≤ Exercise 10.2 Show that χ(G H) χ(G)χ(H): · ≤ Exercise 10.3 Show that Θ(G) χ(G): ≤ So, α(G) Θ(G) χ(G): ≤ ≤ Definition: G is perfect if for all induced sugraphs H of G, α(H) = χ(H); i. e., the chromatic number is equal to the clique number. Theorem 10.4 (Lov´asz) G is perfect iff G is perfect. (This was open under the name \weak perfect graph conjecture.") Corollary 10.5 If G is perfect then Θ(G) = α(G) = χ(G): Exercise 10.6 (a) Kn is perfect. (b) All bipartite graphs are perfect. Exercise 10.7 Prove: If G is bipartite then G is perfect. Do not use Lov´asz'sTheorem (Theorem 10.4). 1 Lecture 10: April 20, 2005 2 The smallest imperfect (not perfect) graph is C5 : α(C5) = 2; χ(C5) = 3: For k 2, C2k+1 imperfect.
    [Show full text]
  • 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).
    [Show full text]
  • Induced Minors and Well-Quasi-Ordering Jaroslaw Blasiok, Marcin Kamiński, Jean-Florent Raymond, Théophile Trunck
    Induced minors and well-quasi-ordering Jaroslaw Blasiok, Marcin Kamiński, Jean-Florent Raymond, Théophile Trunck To cite this version: Jaroslaw Blasiok, Marcin Kamiński, Jean-Florent Raymond, Théophile Trunck. Induced minors and well-quasi-ordering. EuroComb: European Conference on Combinatorics, Graph Theory and Appli- cations, Aug 2015, Bergen, Norway. pp.197-201, 10.1016/j.endm.2015.06.029. lirmm-01349277 HAL Id: lirmm-01349277 https://hal-lirmm.ccsd.cnrs.fr/lirmm-01349277 Submitted on 27 Jul 2016 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. Available online at www.sciencedirect.com www.elsevier.com/locate/endm Induced minors and well-quasi-ordering Jaroslaw Blasiok 1,2 School of Engineering and Applied Sciences, Harvard University, United States Marcin Kami´nski 2 Institute of Computer Science, University of Warsaw, Poland Jean-Florent Raymond 2 Institute of Computer Science, University of Warsaw, Poland, and LIRMM, University of Montpellier, France Th´eophile Trunck 2 LIP, ENS´ de Lyon, France Abstract AgraphH is an induced minor of a graph G if it can be obtained from an induced subgraph of G by contracting edges. Otherwise, G is said to be H-induced minor- free.
    [Show full text]
  • Arxiv:1907.08517V3 [Math.CO] 12 Mar 2021
    RANDOM COGRAPHS: BROWNIAN GRAPHON LIMIT AND ASYMPTOTIC DEGREE DISTRIBUTION FRÉDÉRIQUE BASSINO, MATHILDE BOUVEL, VALENTIN FÉRAY, LUCAS GERIN, MICKAËL MAAZOUN, AND ADELINE PIERROT ABSTRACT. We consider uniform random cographs (either labeled or unlabeled) of large size. Our first main result is the convergence towards a Brownian limiting object in the space of graphons. We then show that the degree of a uniform random vertex in a uniform cograph is of order n, and converges after normalization to the Lebesgue measure on [0; 1]. We finally an- alyze the vertex connectivity (i.e. the minimal number of vertices whose removal disconnects the graph) of random connected cographs, and show that this statistics converges in distribution without renormalization. Unlike for the graphon limit and for the degree of a random vertex, the limiting distribution of the vertex connectivity is different in the labeled and unlabeled settings. Our proofs rely on the classical encoding of cographs via cotrees. We then use mainly combi- natorial arguments, including the symbolic method and singularity analysis. 1. INTRODUCTION 1.1. Motivation. Random graphs are arguably the most studied objects at the interface of com- binatorics and probability theory. One aspect of their study consists in analyzing a uniform random graph of large size n in a prescribed family, e.g. perfect graphs [MY19], planar graphs [Noy14], graphs embeddable in a surface of given genus [DKS17], graphs in subcritical classes [PSW16], hereditary classes [HJS18] or addable classes [MSW06, CP19]. The present paper focuses on uniform random cographs (both in the labeled and unlabeled settings). Cographs were introduced in the seventies by several authors independently, see e.g.
    [Show full text]
  • On Computing Longest Paths in Small Graph Classes
    On Computing Longest Paths in Small Graph Classes Ryuhei Uehara∗ Yushi Uno† July 28, 2005 Abstract The longest path problem is to find a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, few graph classes are known to be solved efficiently for the longest path problem. For a tree, a simple linear time algorithm for the longest path problem is known. We first generalize the algorithm, and show that the longest path problem can be solved efficiently for weighted trees, block graphs, and cacti. We next show that the longest path problem can be solved efficiently on some graph classes that have natural interval representations. Keywords: efficient algorithms, graph classes, longest path problem. 1 Introduction The Hamiltonian path problem is one of the most well known NP-hard problem, and there are numerous applications of the problems [17]. For such an intractable problem, there are two major approaches; approximation algorithms [20, 2, 35] and algorithms with parameterized complexity analyses [15]. In both approaches, we have to change the decision problem to the optimization problem. Therefore the longest path problem is one of the basic problems from the viewpoint of combinatorial optimization. From the practical point of view, it is also very natural approach to try to find a longest path in a given graph, even if it does not have a Hamiltonian path. However, finding a longest path seems to be more difficult than determining whether the given graph has a Hamiltonian path or not.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Discrete Mathematics Rooted Directed Path Graphs Are Leaf Powers
    Discrete Mathematics 310 (2010) 897–910 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Rooted directed path graphs are leaf powers Andreas Brandstädt a, Christian Hundt a, Federico Mancini b, Peter Wagner a a Institut für Informatik, Universität Rostock, D-18051, Germany b Department of Informatics, University of Bergen, N-5020 Bergen, Norway article info a b s t r a c t Article history: Leaf powers are a graph class which has been introduced to model the problem of Received 11 March 2009 reconstructing phylogenetic trees. A graph G D .V ; E/ is called k-leaf power if it admits a Received in revised form 2 September 2009 k-leaf root, i.e., a tree T with leaves V such that uv is an edge in G if and only if the distance Accepted 13 October 2009 between u and v in T is at most k. Moroever, a graph is simply called leaf power if it is a Available online 30 October 2009 k-leaf power for some k 2 N. This paper characterizes leaf powers in terms of their relation to several other known graph classes. It also addresses the problem of deciding whether a Keywords: given graph is a k-leaf power. Leaf powers Leaf roots We show that the class of leaf powers coincides with fixed tolerance NeST graphs, a Graph class inclusions well-known graph class with absolutely different motivations. After this, we provide the Strongly chordal graphs largest currently known proper subclass of leaf powers, i.e, the class of rooted directed path Fixed tolerance NeST graphs graphs.
    [Show full text]
  • Parameterized Algorithms for Recognizing Monopolar and 2
    Parameterized Algorithms for Recognizing Monopolar and 2-Subcolorable Graphs∗ Iyad Kanj1, Christian Komusiewicz2, Manuel Sorge3, and Erik Jan van Leeuwen4 1School of Computing, DePaul University, Chicago, USA, [email protected] 2Institut für Informatik, Friedrich-Schiller-Universität Jena, Germany, [email protected] 3Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Germany, [email protected] 4Max-Planck-Institut für Informatik, Saarland Informatics Campus, Saarbrücken, Germany, [email protected] October 16, 2018 Abstract A graph G is a (ΠA, ΠB)-graph if V (G) can be bipartitioned into A and B such that G[A] satisfies property ΠA and G[B] satisfies property ΠB. The (ΠA, ΠB)-Recognition problem is to recognize whether a given graph is a (ΠA, ΠB )-graph. There are many (ΠA, ΠB)-Recognition problems, in- cluding the recognition problems for bipartite, split, and unipolar graphs. We present efficient algorithms for many cases of (ΠA, ΠB)-Recognition based on a technique which we dub inductive recognition. In particular, we give fixed-parameter algorithms for two NP-hard (ΠA, ΠB)-Recognition problems, Monopolar Recognition and 2-Subcoloring, parameter- ized by the number of maximal cliques in G[A]. We complement our algo- rithmic results with several hardness results for (ΠA, ΠB )-Recognition. arXiv:1702.04322v2 [cs.CC] 4 Jan 2018 1 Introduction A (ΠA, ΠB)-graph, for graph properties ΠA, ΠB , is a graph G = (V, E) for which V admits a partition into two sets A, B such that G[A] satisfies ΠA and G[B] satisfies ΠB. There is an abundance of (ΠA, ΠB )-graph classes, and important ones include bipartite graphs (which admit a partition into two independent ∗A preliminary version of this paper appeared in SWAT 2016, volume 53 of LIPIcs, pages 14:1–14:14.
    [Show full text]
  • Partitioning a Graph Into Disjoint Cliques and a Triangle-Free Graph
    This is a repository copy of Partitioning a graph into disjoint cliques and a triangle-free graph. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/85292/ Version: Accepted Version Article: Abu-Khzam, FN, Feghali, C and Muller, H (2015) Partitioning a graph into disjoint cliques and a triangle-free graph. Discrete Applied Mathematics, 190-19. 1 - 12. ISSN 0166-218X https://doi.org/10.1016/j.dam.2015.03.015 © 2015, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/ Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Partitioning a Graph into Disjoint Cliques and a Triangle-free Graph Faisal N. Abu-Khzam, Carl Feghali, Haiko M¨uller Abstract A graph G =(V, E) is partitionable if there exists a partition {A,B} of V such that A induces a disjoint union of cliques and B induces a triangle- free graph.
    [Show full text]
  • Arxiv:1906.05510V2 [Math.AC]
    BINOMIAL EDGE IDEALS OF COGRAPHS THOMAS KAHLE AND JONAS KRUSEMANN¨ Abstract. We determine the Castelnuovo–Mumford regularity of binomial edge ideals of complement reducible graphs (cographs). For cographs with n vertices the maximum regularity grows as 2n/3. We also bound the regularity by graph theoretic invariants and construct a family of counterexamples to a conjecture of Hibi and Matsuda. 1. Introduction Let G = ([n], E) be a simple undirected graph on the vertex set [n]= {1,...,n}. x1 ··· xn x1 ··· xn Let X = ( y1 ··· yn ) be a generic 2 × n matrix and S = k[ y1 ··· yn ] the polynomial ring whose indeterminates are the entries of X and with coefficients in a field k. The binomial edge ideal of G is JG = hxiyj −yixj : {i, j} ∈ Ei⊆ S, the ideal of 2×2 mi- nors indexed by the edges of the graph. Since their inception in [5, 15], connecting combinatorial properties of G with algebraic properties of JG or S/ JG has been a popular activity. Particular attention has been paid to the minimal free resolution of S/ JG as a standard N-graded S-module [3, 11]. The data of a minimal free res- olution is encoded in its graded Betti numbers βi,j (S/ JG) = dimk Tori(S/ JG, k)j . An interesting invariant is the highest degree appearing in the resolution, the Castelnuovo–Mumford regularity reg(S/ JG) = max{j − i : βij (S/ JG) 6= 0}. It is a complexity measure as low regularity implies favorable properties like vanish- ing of local cohomology. Binomial edge ideals have square-free initial ideals by arXiv:1906.05510v2 [math.AC] 10 Mar 2021 [5, Theorem 2.1] and, using [1], this implies that the extremal Betti numbers and regularity can also be derived from those initial ideals.
    [Show full text]
  • 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.
    [Show full text]