A Survey on Algorithmic Aspects of Modular Decomposition∗
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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. -
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. -
Closest 4-Leaf Power Is Fixed-Parameter Tractable ⋆
Discrete Applied Mathematics, to appear, 2008 Closest 4-Leaf Power is Fixed-Parameter Tractable ⋆ Michael Dom, Jiong Guo, Falk H¨uffner, Rolf Niedermeier Abstract The NP-complete Closest 4-Leaf Power problem asks, given an undirected graph, whether it can be modified by at most r edge insertions or deletions such that it becomes a 4-leaf power. Herein, a 4-leaf power is a graph that can be constructed by considering an unrooted tree—the 4-leaf root—with leaves one-to-one labeled by the graph vertices, where we connect two graph vertices by an edge iff their corresponding leaves are at distance at most 4 in the tree. Complementing previous work on Closest 2-Leaf Power and Closest 3-Leaf Power, we give the first algorithmic result for Closest 4-Leaf Power, showing that Closest 4-Leaf Power is fixed-parameter tractable with respect to the parameter r. Key words: fixed-parameter tractability, graph algorithm, graph modification, graph power, leaf power, forbidden subgraph characterization 1 Introduction Graph powers form a classical concept in graph theory, and the rich literature dates back to the sixties of the previous century (see [5, Sect. 10.6] and [27] for surveys). The k-power of an undirected graph G = (V, E) is the undirected graph Gk = (V, E′) with (u, v) ∈ E′ iff there is a path of length at most k between u and v in G. We say G is the k-root of Gk. It is NP-complete to decide whether a given graph is a 2-power (square) [30]. -
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. -
A Polynomial Kernel for 3-Leaf Power Deletion
A Polynomial Kernel for 3-Leaf Power Deletion Jungho Ahn Department of Mathematical Sciences, KAIST, Daejeon, South Korea Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea [email protected] Eduard Eiben Department of Computer Science, Royal Holloway, University of London, Egham, UK [email protected] O-joung Kwon Department of Mathematics, Incheon National University, South Korea Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea [email protected] Sang-il Oum Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea Department of Mathematical Sciences, KAIST, Daejeon, South Korea [email protected] Abstract For a non-negative integer `, a graph G is an `-leaf power of a tree T if V pGq is equal to the set of leaves of T , and distinct vertices v and w of G are adjacent if and only if the distance between v and w in T is at most `. Given a graph G, 3-Leaf Power Deletion asks whether there is a set S Ď V pGq of size at most k such that GzS is a 3-leaf power of some tree T . We provide a polynomial kernel for this problem. More specifically, we present a polynomial-time algorithm for an input instance pG, kq to output an equivalent instance pG1, k1q such that k1 ¤ k and G1 has at most Opk14q vertices. 2012 ACM Subject Classification Theory of computation Ñ Design and analysis of algorithms; Theory of computation Ñ Graph algorithms analysis; Theory of computation Ñ Parameterized complexity and exact algorithms Keywords and phrases `-leaf power, parameterized algorithms, kernelization Digital Object Identifier 10.4230/LIPIcs.MFCS.2020.5 Related Version A full version of this paper is available at https://arxiv.org/abs/1911.04249. -
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 Leaf Power Recognition Via Embedding Into Graph Products
Parameterized Leaf Power Recognition via Embedding into Graph Products David Eppstein1 Computer Science Department, University of California, Irvine, USA [email protected] Elham Havvaei Computer Science Department, University of California, Irvine, USA [email protected] Abstract The k-leaf power graph G of a tree T is a graph whose vertices are the leaves of T and whose edges connect pairs of leaves at unweighted distance at most k in T . Recognition of the k-leaf power graphs for k ≥ 7 is still an open problem. In this paper, we provide two algorithms for this problem for sparse leaf power graphs. Our results shows that the problem of recognizing these graphs is fixed-parameter tractable when parameterized both by k and by the degeneracy of the given graph. To prove this, we first describe how to embed a leaf root of a leaf power graph into a product of the graph with a cycle graph. We bound the treewidth of the resulting product in terms of k and the degeneracy of G. The first presented algorithm uses methods based on monadic second-order logic (MSO2) to recognize the existence of a leaf power as a subgraph of the graph product. Using the same embedding in the graph product, the second algorithm presents a dynamic programming approach to solve the problem and provide a better dependence on the parameters. Keywords and phrases leaf power, phylogenetic tree, monadic second-order logic, Courcelle’s theorem, strong product of graphs, fixed-parameter tractability, dynamic programming, tree decomposition 1 Introduction Leaf powers are a class of graphs that were introduced in 2002 by Nishimura, Ragde and Thilikos [41], extending the notion of graph powers. -
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. -
Modular Decomposition and Transitive Orientation Ross M
Discrete Mathematics 201 (1999) 189–241 Modular decomposition and transitive orientation Ross M. McConnell a;1, Jeremy P. Spinrad b;∗ a Department of Computer Science and Engineering, University of Colorado at Denver, P.O. Box 173364, Denver, CO 80217-3364, USA b Department of Computer Science, Vanderbilt University, Nashville, TN 37235, USA Abstract A module of an undirected graph is a set X of nodes such for each node x not in X , either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linear-space representation for the modules of a graph, called the modular decomposition. Closely related to modular decomposition is the transitive orientation problem, which is the problem of assigning a direction to each edge of a graph so that the resulting digraph is transitive. A graph is a comparability graph if such an assignment is possible. We give O(n + m) algorithms for modular decomposition and transitive orientation, where n and m are the number of vertices and edges of the graph. This gives linear time bounds for recognizing permutation graphs, maximum clique and minimum vertex coloring on comparability graphs, and other combinatorial problems on comparability graphs and their complements. c 1999 Published by Elsevier Science B.V. All rights reserved 1. Introduction A partial order may be viewed as a transitive directed acyclic graph. A comparabil- ity graph is the graph obtained by ignoring the edge directions of a transitive directed acyclic graph. It is well known that every partial order is the intersection of a set of total (linear) orders [10]. -
Polynomial Kernels for 3-Leaf Power Graph Modification Problems
Polynomial kernels for 3-leaf power graph modification problems∗ S. Bessy C. Paul A. Perez CNRS, LIRMM, Universit´eMontpellier 2, France † September 17, 2008 Abstract A graph G = (V, E) is a 3-leaf power iff there exists a tree T whose leaves are V and such that (u, v) ∈ E iff u and v are at distance at most 3 in T . The 3-leaf power graph edge modification problems, i.e. edition (also known as the closest 3-leaf power), completion and edge-deletion, are FTP when parameterized by the size of the edge set modification. However polynomial kernel was known for none of these three problems. For each of them, we provide cubic kernels that can be computed in linear time for each of these problems. We thereby answer an open problem first mentioned by Dom, Guo, H¨uffner and Niedermeier [6]. arXiv:0809.2858v2 [cs.DM] 17 Sep 2008 ∗Research supported by the ANR anr-blan-0148-06project ”Graph Decomposition on Algorithm (GRAAL)” †e-mail: [email protected], [email protected], [email protected] 1 Introduction An edge modification problem aims at changing the edge set of an input graph G = (V, E) in order to get a certain property Π satisfied (see [17] for a recent study). Edge modification prob- lems cover a broad range of graph optimization problems among which completion problems (e.g. minimum chordal completion), edition problems (e.g. cluster edition) and edge deletion problems (e.g. maximum planar subgraph). For completion problems, the set F of modified edges is constraint to be disjoint from E, whereas for edge deletion problems F has to be a subset of E. -
Polynomial Kernels for 3-Leaf Power Graph Modification Problems
Polynomial kernels for 3-leaf power graph modification problems∗ S. Bessy C. Paul A. Perez CNRS, LIRMM, Universit´eMontpellier 2, France † October 26, 2018 Abstract A graph G = (V, E) is a 3-leaf power iff there exists a tree T whose leaves are V and such that (u, v) ∈ E iff u and v are at distance at most 3 in T . The 3-leaf power graph edge modification problems, i.e. edition (also known as the closest 3-leaf power), completion and edge-deletion, are FTP when parameterized by the size of the edge set modification. However polynomial kernel was known for none of these three problems. For each of them, we provide cubic kernels that can be computed in linear time for each of these problems. We thereby answer an open problem first mentioned by Dom, Guo, H¨uffner and Niedermeier [6]. arXiv:0809.2858v2 [cs.DM] 17 Sep 2008 ∗Research supported by the ANR anr-blan-0148-06project ”Graph Decomposition on Algorithm (GRAAL)” †e-mail: [email protected], [email protected], [email protected] 1 Introduction An edge modification problem aims at changing the edge set of an input graph G = (V, E) in order to get a certain property Π satisfied (see [17] for a recent study). Edge modification prob- lems cover a broad range of graph optimization problems among which completion problems (e.g. minimum chordal completion), edition problems (e.g. cluster edition) and edge deletion problems (e.g. maximum planar subgraph). For completion problems, the set F of modified edges is constraint to be disjoint from E, whereas for edge deletion problems F has to be a subset of E.