Algorithmics of Modular Decomposition
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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. -
5Th Lecture : Modular Decomposition MPRI 2013–2014 Schedule A
5th Lecture : Modular decomposition MPRI 2013–2014 5th Lecture : Modular decomposition MPRI 2013–2014 Schedule 5th Lecture : Modular decomposition Introduction MPRI 2013–2014 Graph searches Michel Habib Applications of LBFS on structured graph classes [email protected] http://www.liafa.univ-Paris-Diderot.fr/~habib Chordal graphs Cograph recognition Sophie Germain, 22 octobre 2013 A nice conjecture 5th Lecture : Modular decomposition MPRI 2013–2014 5th Lecture : Modular decomposition MPRI 2013–2014 Introduction A hierarchy of graph models 1. Undirected graphs (graphes non orient´es) 2. Tournaments (Tournois), sometimes 2-circuits are allowed. 3. Signed graphs (Graphes sign´es) each edge is labelled + or - (for example friend or enemy) Examen le mardi 26 novembre de 9h `a12h 4. Oriented graphs (Graphes orient´es), each edge is given a Salle habituelle unique direction (no 2-circuits) An interesting subclass are the DAG Directed Acyclic Graphs (graphes sans circuit), for which the transitive closure is a partial order (ordre partiel) 5. Partial orders and comparability graphs an intersting particular case. Duality comparability – cocomparability (graphes de comparabilit´e– graphes d’incomparabilit´e) 6. Directed graphs or digraphs (Graphes dirig´es) 5th Lecture : Modular decomposition MPRI 2013–2014 5th Lecture : Modular decomposition MPRI 2013–2014 Introduction Introduction The problem has to be defined in each model and sometimes it could be hard. ◮ What is the right notion for a coloration in a directed graph ? For partial orders, comparability graphs or uncomparability graphs ◮ No directed cycle unicolored, seems to be the good one. the independant set and maximum clique problems are polynomial. ◮ It took 20 years to find the right notion of oriented matro¨ıd ◮ What is the right notion of treewidth for directed graphs ? ◮ Still an open question. -
Split Decomposition and Graph-Labelled Trees
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Applied Mathematics 160 (2012) 708–733 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Split decomposition and graph-labelled trees: Characterizations and fully dynamic algorithms for totally decomposable graphsI Emeric Gioan, Christophe Paul ∗ CNRS - LIRMM, Université de Montpellier 2, France article info a b s t r a c t Article history: In this paper, we revisit the split decomposition of graphs and give new combinatorial and Received 20 May 2010 algorithmic results for the class of totally decomposable graphs, also known as the distance Received in revised form 3 February 2011 hereditary graphs, and for two non-trivial subclasses, namely the cographs and the 3-leaf Accepted 23 May 2011 power graphs. Precisely, we give structural and incremental characterizations, leading to Available online 23 July 2011 optimal fully dynamic recognition algorithms for vertex and edge modifications, for each of these classes. These results rely on the new combinatorial framework of graph-labelled Keywords: trees used to represent the split decomposition of general graphs (and also the modular Split decomposition Distance hereditary graphs decomposition). The point of the paper is to use bijections between the aforementioned Fully dynamic algorithms graph classes and graph-labelled trees whose nodes are labelled by cliques and stars. We mention that this bijective viewpoint yields directly an intersection model for the class of distance hereditary graphs. ' 2012 Published by Elsevier B.V. 1. Introduction The 1-join composition and its complementary operation, the split decomposition, range among the classical operations in graph theory. -
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. -
Applications of Lexicographic Breadth-First Search to Modular Decomposition, Split Decomposition, and Circle Graphs by Marc Tedd
Applications of Lexicographic Breadth-First Search to Modular Decomposition, Split Decomposition, and Circle Graphs by Marc Tedder A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Computer Science University of Toronto Copyright c 2011 by Marc Tedder Abstract Applications of Lexicographic Breadth-First Search to Modular Decomposition, Split Decomposition, and Circle Graphs Marc Tedder Doctor of Philosophy Graduate Department of Computer Science University of Toronto 2011 This thesis presents the first sub-quadratic circle graph recognition algorithm, and develops im- proved algorithms for two important hierarchical decomposition schemes: modular decomposition and split decomposition. The modular decomposition algorithm results from unifying two dif- ferent approaches previously employed to solve the problem: divide-and-conquer and factorizing permutations. It runs in linear-time, and is straightforward in its understanding, correctness, and implementation. It merely requires a collection of trees and simple traversals of these trees. The split-decomposition algorithm is similar in being straightforward in its understanding and correctness. An efficient implementation of the algorithm is described that uses the union-find data-structure. A novel charging argument is used to prove the running-time. The algorithm is the first to use the recent reformulation of split decomposition in terms of graph-labelled trees. This facilitates its extension to circle graph recognition. In particular, it allows us to efficiently apply a new lexicographic breadth-first search characterization of circle graphs developed in the thesis. Lexicographic breadth-first search is additionally responsible for the efficiency of the split decom- position algorithm, and contributes to the simplicity of the modular decomposition algorithm. -
Treewidth and Pathwidth of Permutation Graphs
Treewidth and pathwidth of permutation graphs Citation for published version (APA): Bodlaender, H. L., Kloks, A. J. J., & Kratsch, D. (1992). Treewidth and pathwidth of permutation graphs. (Universiteit Utrecht. UU-CS, Department of Computer Science; Vol. 9230). Utrecht University. Document status and date: Published: 01/01/1992 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. -
Classes of Perfect Graphs
This paper appeared in: Discrete Mathematics 306 (2006), 2529-2571 Classes of Perfect Graphs Stefan Hougardy Humboldt-Universit¨atzu Berlin Institut f¨urInformatik 10099 Berlin, Germany [email protected] February 28, 2003 revised October 2003, February 2005, and July 2007 Abstract. The Strong Perfect Graph Conjecture, suggested by Claude Berge in 1960, had a major impact on the development of graph theory over the last forty years. It has led to the definitions and study of many new classes of graphs for which the Strong Perfect Graph Conjecture has been verified. Powerful concepts and methods have been developed to prove the Strong Perfect Graph Conjecture for these special cases. In this paper we survey 120 of these classes, list their fundamental algorithmic properties and present all known relations between them. 1 Introduction A graph is called perfect if the chromatic number and the clique number have the same value for each of its induced subgraphs. The notion of perfect graphs was introduced by Berge [6] in 1960. He also conjectured that a graph is perfect if and only if it contains, as an induced subgraph, neither an odd cycle of length at least five nor its complement. This conjecture became known as the Strong Perfect Graph Conjecture and attempts to prove it contributed much to the developement of graph theory in the past forty years. The methods developed and the results proved have their uses also outside the area of perfect graphs. The theory of antiblocking polyhedra developed by Fulkerson [37], and the theory of modular decomposition (which has its origins in a paper of Gallai [39]) are two such examples. -
The Use of a Pruned Modular Decomposition for Maximum Matching Algorithms on Some Graph Classes
The Use of a Pruned Modular Decomposition for Maximum Matching Algorithms on Some Graph Classes Guillaume Ducoffe ICI – National Institute for Research and Development in Informatics, Bucharest, Romania The Research Institute of the University of Bucharest ICUB, Bucharest, Romania guillaume.ducoff[email protected] Alexandru Popa University of Bucharest, Bucharest, Romania ICI – National Institute for Research and Development in Informatics, Bucharest, Romania [email protected] Abstract We address the following general question: given a graph class C on which we can solve Maximum Matching in (quasi) linear time, does the same hold true for the class of graphs that can be modularly decomposed into C? As a way to answer this question for distance-hereditary graphs and some other superclasses of cographs, we study the combined effect of modular decomposition with a pruning process over the quotient subgraphs. We remove sequentially from all such subgraphs their so-called one-vertex extensions (i.e., pendant, anti-pendant, twin, universal and isolated vertices). Doing so, we obtain a “pruned modular decomposition”, that can be computed in quasi linear time. Our main result is that if all the pruned quotient subgraphs have bounded order then a maximum matching can be computed in linear time. The latter result strictly extends a recent framework in (Coudert et al., SODA’18). Our work is the first to explain why the existence of some nice ordering over the modules of a graph, instead of just over its vertices, can help to speed up the computation of maximum matchings on some graph classes. 2012 ACM Subject Classification Mathematics of computing → Graph theory, Theory of com- putation → Design and analysis of algorithms Keywords and phrases maximum matching, FPT in P, modular decomposition, pruned graphs, one-vertex extensions, P4-structure Digital Object Identifier 10.4230/LIPIcs.ISAAC.2018.6 Related Version A full version of the paper is available at [14], https://arxiv.org/abs/1804. -
Parameterized Complexity of Induced Graph Matching on Claw-Free Graphs
Parameterized Complexity of Induced Graph Matching on Claw-Free Graphs∗ Danny Hermelin† Matthias Mnich‡ Erik Jan van Leeuwen§ Abstract The Induced Graph Matching problem asks to find k disjoint induced subgraphs isomorphic to a given graph H in a given graph G such that there are no edges between vertices of different subgraphs. This problem generalizes the classical Independent Set and Induced Matching problems, among several other problems. We show that In- duced Graph Matching is fixed-parameter tractable in k on claw-free graphs when H is a fixed connected graph, and even admits a polynomial kernel when H is a complete graph. Both results rely on a new, strong, and generic algorithmic structure theorem for claw-free graphs. Complementing the above positive results, we prove W[1]-hardness of Induced Graph Matching on graphs excluding K1,4 as an induced subgraph, for any fixed complete graph H. In particular, we show that Independent Set is W[1]-hard on K1,4-free graphs. Finally, we consider the complexity of Induced Graph Matching on a large subclass of claw-free graphs, namely on proper circular-arc graphs. We show that the problem is either polynomial-time solvable or NP-complete, depending on the connectivity of H and the structure of G. 1 Introduction A graph is claw-free if no vertex in the graph has three pairwise nonadjacent neighbors, i.e. if it does not contain a copy of K1,3 as an induced subgraph. The class of claw-free graphs contains several well-studied graph classes such as line graphs, unit interval graphs, de Bruijn graphs, the complements of triangle-free graphs, and graphs of several polyhedra and polytopes. -
Permutations and Permutation Graphs
Permutations and permutation graphs Robert Brignall Schloss Dagstuhl, 8 November 2018 1 / 25 Permutations and permutation graphs 4 1 2 6 3 8 5 7 • Permutation p = p(1) ··· p(n) • Inversion graph Gp: for i < j, ij 2 E(Gp) iff p(i) > p(j). • Note: n ··· 21 becomes Kn. • Permutation graph = can be made from a permutation 2 / 25 Permutations and permutation graphs 4 1 2 6 3 8 5 7 • Permutation p = p(1) ··· p(n) • Inversion graph Gp: for i < j, ij 2 E(Gp) iff p(i) > p(j). • Note: n ··· 21 becomes Kn. • Permutation graph = can be made from a permutation 2 / 25 Ordering permutations: containment 1 3 5 2 4 < 4 1 2 6 3 8 5 7 • ‘Classical’ pattern containment: s ≤ p. • Translates to induced subgraphs: Gs ≤ind Gp. • Permutation class: a downset: p 2 C and s ≤ p implies s 2 C. • Avoidance: minimal forbidden permutation characterisation: C = Av(B) = fp : b 6≤ p for all b 2 Bg. 3 / 25 Ordering permutations: containment 1 3 5 2 4 < 4 1 2 6 3 8 5 7 • ‘Classical’ pattern containment: s ≤ p. • Translates to induced subgraphs: Gs ≤ind Gp. • Permutation class: a downset: p 2 C and s ≤ p implies s 2 C. • Avoidance: minimal forbidden permutation characterisation: C = Av(B) = fp : b 6≤ p for all b 2 Bg. 3 / 25 Ordering permutations: containment 1 3 5 2 4 < 4 1 2 6 3 8 5 7 • ‘Classical’ pattern containment: s ≤ p. • Translates to induced subgraphs: Gs ≤ind Gp. • Permutation class: a downset: p 2 C and s ≤ p implies s 2 C. -
Fully Dynamic Algorithm for Recognition and Modular Decomposition of Permutation Graphs
Algorithmica DOI 10.1007/s00453-008-9273-0 Fully Dynamic Algorithm for Recognition and Modular Decomposition of Permutation Graphs Christophe Crespelle · Christophe Paul Received: 5 June 2006 / Accepted: 22 December 2008 © Springer Science+Business Media, LLC 2009 Abstract This paper considers the problem of maintaining a compact representa- tion (O(n) space) of permutation graphs under vertex and edge modifications (inser- tion or deletion). That representation allows us to answer adjacency queries in O(1) time. The approach is based on a fully dynamic modular decomposition algorithm for permutation graphs that works in O(n) time per edge and vertex modification. We thereby obtain a fully dynamic algorithm for the recognition of permutation graphs. Keywords Dynamic algorithms · Permutation graphs · Modular decomposition 1 Introduction Finding efficient graph representations is a central question of algorithmic graph the- ory. How to store a graph to make its manipulation easier? Compact graph represen- tations often rest on the combinatorial structures of the considered graphs (see for example [22]). Thereby interesting graph encodings can be found when restricted to special graph classes. This paper deals with dynamic graphs and considers the dy- namic recognition and representation problem (see e.g. [21]), which, for a family F of graphs, aims to maintain a characteristic representation of dynamically changing graphs as long as the modified graph belongs to F. The input of the problem is a graph G ∈ F with its representation and a series of modifications. Any modification This paper is a full version of the extended abstract appeared in [5]. C. Crespelle () Université de Montpellier 2, LIRMM, Montpellier, France e-mail: [email protected] C. -
Permutation Graphs --- an Introduction
Introduction Optimization problems Recognition Encoding all realizers Conclusion Permutation graphs | an introduction Ioan Todinca LIFO - Universit´ed'Orl´eans Algorithms and permutations, february 2012 1/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion Permutation graphs 1 23 4 2 3 4 1 5 6 5 6 • Optimisation algorithms use, as input, the intersection model (realizer) • Recognition algorithms output the intersection(s) model(s) 2/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion Plan of the talk 1. Relationship with other graph classes 2. Optimisation problems : MaxIndependentSet/MaxClique/Coloring ; Treewidth 3. Recognition algorithm 4. Encoding all realizers via modular decomposition 5. Conclusion 3/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion Definition and basic properties 2 1 6 5 4 3 1 23 4 2 3 4 1 5 6 5 6 1 5 6 3 2 4 • Realizer:( π1; π2) • One can "reverse" the realizer upside-down or right-left : (π1; π2) ∼ (π2; π1) ∼ (π1; π2) ∼ (π2; π1) • Complements of permutation graphs are permutation graphs. ! Reverse the ordering of the bottoms of the segments : (π1; π2) ! (π1; π2) 4/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion Definition and basic properties 2 1 6 5 4 3 1 23 4 5 6 1 5 6 3 2 4 • Realizer:( π1; π2) • One can "reverse" the realizer upside-down or right-left : (π1; π2) ∼ (π2; π1) ∼ (π1; π2) ∼ (π2; π1) • Complements of permutation graphs are permutation graphs. ! Reverse the ordering of the bottoms of the segments : (π1; π2) ! (π1; π2) 4/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion Definition and basic properties 2 1 6 5 4 3 1 23 4 5 6 4 2 3 6 5 1 • Realizer:( π1; π2) • One can "reverse" the realizer upside-down or right-left : (π1; π2) ∼ (π2; π1) ∼ (π1; π2) ∼ (π2; π1) • Complements of permutation graphs are permutation graphs.