Clustering Powers of Sparse Graphs
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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. -
Sub-Coloring and Hypo-Coloring Interval Graphs⋆
Sub-coloring and Hypo-coloring Interval Graphs? Rajiv Gandhi1, Bradford Greening, Jr.1, Sriram Pemmaraju2, and Rajiv Raman3 1 Department of Computer Science, Rutgers University-Camden, Camden, NJ 08102. E-mail: [email protected]. 2 Department of Computer Science, University of Iowa, Iowa City, Iowa 52242. E-mail: [email protected]. 3 Max-Planck Institute for Informatik, Saarbr¨ucken, Germany. E-mail: [email protected]. Abstract. In this paper, we study the sub-coloring and hypo-coloring problems on interval graphs. These problems have applications in job scheduling and distributed computing and can be used as “subroutines” for other combinatorial optimization problems. In the sub-coloring problem, given a graph G, we want to partition the vertices of G into minimum number of sub-color classes, where each sub-color class induces a union of disjoint cliques in G. In the hypo-coloring problem, given a graph G, and integral weights on vertices, we want to find a partition of the vertices of G into sub-color classes such that the sum of the weights of the heaviest cliques in each sub-color class is minimized. We present a “forbidden subgraph” characterization of graphs with sub-chromatic number k and use this to derive a a 3-approximation algorithm for sub-coloring interval graphs. For the hypo-coloring problem on interval graphs, we first show that it is NP-complete and then via reduction to the max-coloring problem, show how to obtain an O(log n)-approximation algorithm for it. 1 Introduction Given a graph G = (V, E), a k-sub-coloring of G is a partition of V into sub-color classes V1,V2,...,Vk; a subset Vi ⊆ V is called a sub-color class if it induces a union of disjoint cliques in G. -
Layouts of Expander Graphs
CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2016, Article 1, pages 1–21 http://cjtcs.cs.uchicago.edu/ Layouts of Expander Graphs † ‡ Vida Dujmovic´∗ Anastasios Sidiropoulos David R. Wood Received January 20, 2015; Revised November 26, 2015, and on January 3, 2016; Published January 14, 2016 Abstract: Bourgain and Yehudayoff recently constructed O(1)-monotone bipartite ex- panders. By combining this result with a generalisation of the unraveling method of Kannan, we construct 3-monotone bipartite expanders, which is best possible. We then show that the same graphs admit 3-page book embeddings, 2-queue layouts, 4-track layouts, and have simple thickness 2. All these results are best possible. Key words and phrases: expander graph, monotone layout, book embedding, stack layout, queue layout, track layout, thickness 1 Introduction Expanders are classes of highly connected graphs that are of fundamental importance in graph theory, with numerous applications, especially in theoretical computer science [31]. While the literature contains various definitions of expanders, this paper focuses on bipartite expanders. For e (0;1], a bipartite 2 graph G with bipartition V(G) = A B is a bipartite e-expander if A = B and N(S) > (1 + e) S for A [ j j j j j j j j every subset S A with S j j . Here N(S) is the set of vertices adjacent to some vertex in S. An infinite ⊂ j j 6 2 family of bipartite e-expanders, for some fixed e > 0, is called an infinite family of bipartite expanders. There has been much research on constructing and proving the existence of expanders with various desirable properties. -
On Star and Caterpillar Arboricity
Discrete Mathematics 309 (2009) 3694–3702 www.elsevier.com/locate/disc On star and caterpillar arboricity Daniel Gonc¸alvesa,∗, Pascal Ochemb a LIRMM UMR 5506, CNRS, Universite´ Montpelier 2, 161 rue Ada, 34392 Montpellier Cedex 5, France b LRI UMR 8623, CNRS, Universite´ Paris-Sud, Batˆ 490, 91405 Orsay Cedex, France Received 31 October 2005; accepted 18 January 2008 Available online 10 March 2008 Abstract We give new bounds on the star arboricity and the caterpillar arboricity of planar graphs with given girth. One of them answers an open problem of Gyarf´ as´ and West: there exist planar graphs with track number 4. We also provide new NP-complete problems. c 2008 Elsevier B.V. All rights reserved. Keywords: NP-completeness; Partitioning problems; Edge coloring 1. Introduction Many graph parameters in the literature are defined as the minimum size of a partition of the edges of the graph such that each part induces a graph of a given class C. The most common is the chromatic index χ 0.G/, in this case C is the class of graphs with maximum degree one. Vizing [18] proved that χ 0.G/ either equals ∆.G/ or ∆.G/ C 1, where ∆.G/ denotes the maximum degree of G. Deciding whether χ 0.G/ D 3 is shown to be NP-complete for general graphs in [13]. The arboricity a.G/ is another well studied parameter, for which C is the class of forests. In [15], Nash-Williams proved that: jE.H/j a.G/ D max (1) H⊆G jV .H/j − 1 with the maximum being over all the subgraphs H D .E.H/; V .H// of G. -
Characterising Bounded Expansion by Neighbourhood Complexity
Characterising Bounded Expansion by Neighbourhood Complexity Felix Reidl Fernando S´anchez Villaamil Konstantinos Stavropoulos RWTH Aachen University freidl,fernando.sanchez,[email protected]. Abstract We show that a graph class G has bounded expansion if and only if it has bounded r-neighbourhood complexity, i.e. for any vertex set X of any subgraph H of G ∈ G, the number of subsets of X which are exact r-neighbourhoods of vertices of H on X is linear in the size of X. This is established by bounding the r-neighbourhood complexity of a graph in terms of both its r-centred colouring number and its weak r-colouring number, which provide known characterisations to the property of bounded expansion. 1 Introduction Graph classes of bounded expansion (and their further generalisation, nowhere dense classes) have been introduced by Neˇsetˇriland Ossona de Mendez [23, 24, 25] as a general model of structurally sparse graph classes. They include and generalise many other natural sparse graph classes, among them all classes of bounded degree, classes of bounded genus, and classes defined by arXiv:1603.09532v2 [cs.DM] 2 Nov 2016 excluded (topological) minors. Nowhere dense classes even include classes that locally exclude a minor, which in turn generalises graphs with locally bounded treewidth. The appeal of this notion and its applications stems from the fact that bounded expansion has turned out to be a very robust property of graph classes with various seemingly unrelated characterisations (see [19, 25]). These include characterisations through the density of shallow minors [23], quasi-wideness [3] low treedepth colourings [23], and generalised colouring numbers [31]. -
Lecture Beyond Planar Graphs 1 Overview 2 Preliminaries
Approximation Algorithms Workshop June 17, 2011, Princeton Lecture Beyond Planar Graphs Erik Demaine Scribe: Siamak Tazari 1 Overview In the last lecture we learned about approximation schemes in planar graphs. In this lecture we go beyond planar graphs and consider bounded-genus graphs, and more generally, minor-closed graph classes. In particular, we discuss the framework of bidimensionality that can be used to obtain approximation schemes for many problems in minor-closed graph classes. Afterwards, we consider decomposition techniques that can be applied to other types of problems. The main motivation behind this line of work is that some networks are not planar but conform to some generalization of the concept of planarity, like being embeddable on a surface of bounded genus. Furthermore, by Kuratowski's famous theorem [Kur30], planarity can be characterized be excluding K5 and K3;3 as minors. Hence, it seems natural to study further classes with excluded minors as a generalization of planar graphs. One of the goals of this research is to identify the largest classes of graphs for which we can still ob- tain good and efficient approximations, i.e. to find out how far beyond planarity we can go. Natural candidate classes are graphs of bounded genus, graphs excluding a fixed minor, powers thereof, but also further generalizations, like odd-minor-free graphs, graphs of bounded expansion and nowhere dense classes of graphs. Another objective is to derive general approximation frameworks, two of which we are going to discuss here: bidimensionality and contraction decomposition. The main tools we are going to use come on one hand from graph structure theory and on the other hand from algorithmic results. -
On Quasi-Planar Graphs: Clique-Width and Logical Description
On quasi-planar graphs : clique-width and logical description Bruno Courcelle To cite this version: Bruno Courcelle. On quasi-planar graphs : clique-width and logical description. Discrete Applied Mathematics, Elsevier, 2018, 10.1016/j.dam.2018.07.022. hal-01735820v2 HAL Id: hal-01735820 https://hal.archives-ouvertes.fr/hal-01735820v2 Submitted on 3 Mar 2020 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. On quasi-planar graphs : clique-width and logical description. Bruno Courcelle Labri, CNRS and Bordeaux University∗ 33405 Talence, France email: [email protected] June 19, 2018 Abstract Motivated by the construction of FPT graph algorithms parameterized by clique-width or tree-width, we study graph classes for which tree- width and clique-width are linearly related. This is the case for all graph classes of bounded expansion, but in view of concrete applications, we want to have "small" constants in the comparisons between these width parameters. We focus our attention on graphs that can be drawn in the plane with limited edge crossings, for an example, at most p crossings for each edge. These graphs are called p-planar. We consider a more general situation where the graph of edge crossings must belong to a fixed class of graphs . -
Characterisations and Examples of Graph Classes with Bounded Expansion Jaroslav Nešetřil A, Patrice Ossona De Mendez B, David R
European Journal of Combinatorics ( ) – Contents lists available at SciVerse ScienceDirect European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc Characterisations and examples of graph classes with bounded expansion Jaroslav Nešetřil a, Patrice Ossona de Mendez b, David R. Wood c a Department of Applied Mathematics, and Institute for Theoretical Computer Science, Charles University, Prague, Czech Republic b Centre d'Analyse et de Mathématique Sociales, Centre National de la Recherche Scientifique, and École des Hautes Études en Sciences Sociales, Paris, France c Department of Mathematics and Statistics, The University of Melbourne, Melbourne, Australia article info a b s t r a c t Article history: Classes with bounded expansion, which generalise classes that Available online xxxx exclude a topological minor, have recently been introduced by Nešetřil and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Several linear-time algorithms are known for bounded expansion classes (such as subgraph isomorphism testing), and they allow restricted homomorphism dualities, amongst other desirable properties. In this paper, we establish two new characterisations of bounded expansion classes, one in terms of so-called topological parameters and the other in terms of controlling dense parts. The latter characterisation is then used to show that the notion of bounded expansion is compatible with the Erdös–Rényi model of random graphs with constant average degree. In particular, we prove that for every fixed d > 0, there exists a class with bounded expansion, such that a random graph of order n and edge probability d=n asymptotically almost surely belongs to the class. -
An Annotated Bibliography on 1-Planarity
An annotated bibliography on 1-planarity Stephen G. Kobourov1, Giuseppe Liotta2, Fabrizio Montecchiani2 1Dep. of Computer Science, University of Arizona, Tucson, USA [email protected] 2Dip. di Ingegneria, Universit`adegli Studi di Perugia, Perugia, Italy fgiuseppe.liotta,[email protected] July 21, 2017 Abstract The notion of 1-planarity is among the most natural and most studied generalizations of graph planarity. A graph is 1-planar if it has an embedding where each edge is crossed by at most another edge. The study of 1-planar graphs dates back to more than fifty years ago and, recently, it has driven increasing attention in the areas of graph theory, graph algorithms, graph drawing, and computational geometry. This annotated bibliography aims to provide a guiding reference to researchers who want to have an overview of the large body of literature about 1-planar graphs. It reviews the current literature covering various research streams about 1-planarity, such as characterization and recognition, combinatorial properties, and geometric representations. As an additional contribution, we offer a list of open problems on 1-planar graphs. 1 Introduction Relational data sets, containing a set of objects and relations between them, are commonly modeled by graphs, with the objects as the vertices and the relations as the edges. A great deal is known about the structure and properties of special types of graphs, in particular planar graphs. The class of planar graphs is fundamental for both Graph Theory and Graph Algorithms, and is extensively studied. Many structural properties of planar graphs are known and these properties can be used in the development of efficient algorithms for planar graphs, even where the more general problem is NP-hard [11]. -
Logic, Graphs, and Algorithms
Electronic Colloquium on Computational Complexity, Report No. 91 (2007) Logic, Graphs, and Algorithms Martin Grohe Humboldt-Universit¨at zu Berlin September 10, 2007 Abstract Algorithmic meta theorems are algorithmic results that apply to whole families of combinatorial problems, instead of just specific problems. These families are usually defined in terms of logic and graph theory. An archetypal algorithmic meta theorem is Courcelle’s Theorem [9], which states that all graph properties definable in monadic second-order logic can be decided in linear time on graphs of bounded tree width. This article is an introduction into the theory underlying such meta theorems and a survey of the most important results in this area. 1 Introduction In 1990, Courcelle [9] proved a fundamental theorem stating that graph properties definable in monadic second-order logic can be decided in linear time on graphs of bounded tree width. This is the first in a series of algorithmic meta theorems. More recent examples of such meta theorems state that all first-order definable properties of planar graphs can be decided in linear time [43] and that all first-order definable optimisation problems on classes of graphs with excluded minors can be approximated in polynomial time to any given approximation ratio [19]. The term “meta theorem” refers to the fact that these results do not describe algorithms for specific problems, but for whole families of problems, whose definition typically has a logical and a structural (usually graph theoretical) component. For example, Courcelle’s Theorem is about monadic second-order logic on graphs of bounded tree width. -
Colouring and Covering Nowhere Dense Graphs∗
COLOURING AND COVERING NOWHERE DENSE GRAPHS∗ MARTIN GROHE† , STEPHAN KREUTZER‡ , ROMAN RABINOVICH§ , SEBASTIAN SIEBERTZ¶, AND KONSTANTINOS STAVROPOULOSk Abstract. In [12] it was shown that nowhere dense classes of graphs admit sparse neighbourhood covers of small degree. We show that a monotone graph class admits sparse neighbourhood covers if and only if it is nowhere dense. The existence of such covers for nowhere dense classes is established through bounds on so-called weak colouring numbers. The core results of this paper are various lower and upper bounds on the weak colouring numbers and other, closely related generalised colouring numbers. We prove tight bounds for these numbers on graphs of bounded treewidth. We clarify and tighten the relation between the density of shallow minors and the various generalised colouring numbers. These upper bounds are complemented by new, stronger exponential lower bounds on the weak and strong colouring numbers, and by super-polynomial lower bounds on the weak colouring numbers on classes of polynomial expansion. Finally, we show that computing weak r-colouring numbers is NP-complete for all r ≥ 3. 1. Introduction. Nowhere dense classes of graphs have been introduced by Nešetřil and Ossona de Mendez [18, 20] as a general model of uniformly sparse graph classes. They include and generalise many other natural sparse graph classes, among them all classes of bounded degree, classes of bounded genus, classes defined by excluded (topological) minors, and classes of bounded expansion. It has been demonstrated in several papers, e.g., [2, 8, 12, 16, 18] that nowhere dense graph classes have nice algorithmic properties; many problems that are hard in general can be solved (more) efficiently on nowhere dense graph classes. -
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.