Nowhere Dense Classes Of Graphs
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Nowhere Dense Classes Of Graphs
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Author : Siebertz, Sebastian
language : en
Publisher: Universitätsverlag der TU Berlin
Release Date : 2016-05-24
Nowhere Dense Classes Of Graphs written by Siebertz, Sebastian and has been published by Universitätsverlag der TU Berlin this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-05-24 with Computers categories.
We show that every first-order property of graphs can be decided in almost linear time on every nowhere dense class of graphs. For graph classes closed under taking subgraphs, our result is optimal (under a standard complexity theoretic assumption): it was known before that for all classes C of graphs closed under taking subgraphs, if deciding first-order properties of graphs in C is fixed-parameter tractable, parameterized by the length of the input formula, then C must be nowhere dense. Nowhere dense graph classes form a large variety of classes of sparse graphs including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion. For our proof, we provide two new characterisations of nowhere dense classes of graphs. The first characterisation is in terms of a game, which explains the local structure of graphs from nowhere dense classes. The second characterisation is by the existence of sparse neighbourhood covers. On the logical side, we prove a rank-preserving version of Gaifman's locality theorem. The characterisation by neighbourhood covers is based on a characterisation of nowhere dense classes by generalised colouring numbers. We show several new bounds for the generalised colouring numbers on restricted graph classes, such as for proper minor closed classes and for planar graphs. Finally, we study the parameterized complexity of the first-order model-checking problem on structures where an ordering is available to be used in formulas. We show that first-order logic on ordered structures as well as on structures with a successor relation is essentially intractable on nearly all interesting classes. On the other hand, we show that the model-checking problem of order-invariant monadic second-order logic is tractable essentially on the same classes as plain monadic second-order logic and that the model-checking problem for successor-invariant first-order logic is tractable on planar graphs. Wir zeigen, dass jede Eigenschaft von Graphen aus einer nowhere dense Klasse von Graphen, die in der Präadikatenlogik formuliert werden kann, in fast linearer Zeit entschieden werden kann. Dieses Ergebnis ist optimal für Klassen von Graphen, die unter Subgraphen abgeschlossen sind (unter einer Standardannahme aus der Komplexitätstheorie). Um den obigen Satz zu beweisen, führen wir zwei neue Charakterisierungen von nowhere dense Klassen von Graphen ein. Zunächst charakterisieren wir solche Klassen durch ein Spiel, das die lokalen Eigenschaften von Graphen beschreibt. Weiter zeigen wir, dass eine Klasse, die unter Subgraphen abgeschlossen ist, genau dann nowhere dense ist, wenn alle lokalen Nachbarschaften von Graphen der Klasse dünn überdeckt werden können. Weiterhin beweisen wir eine erweiterte Version von Gaifman's Lokalitätssatz für die Prädikatenlogik, der eine Übersetzung von Formeln in lokale Formeln des gleichen Ranges erlaubt. In Kombination erlauben diese neuen Charakterisierungen einen effizienten, rekursiven Lösungsansatz für das Model-Checking Problem der Prädikatenlogik. Die Charakterisierung der nowhere dense Graphklassen durch die oben beschriebenen Überdeckungen basiert auf einer bekannten Charakterisierung durch verallgemeinerte Färbungszahlen. Unser Studium dieser Zahlen führt zu neuen, verbesserten Schranken für die verallgemeinerten Färbungszahlen von nowhere dense Klassen von Graphen, insbesondere für einige wichtige Subklassen, z. B. für Klassen mit ausgeschlossenen Minoren und für planare Graphen. Zuletzt untersuchen wir, welche Auswirkungen eine Erweiterung der Logik durch Ordnungs- bzw. Nachfolgerrelationen auf die Komplexität des Model-Checking Problems hat. Wir zeigen, dass das Problem auf fast allen interessanten Klassen nicht effizient gelöst werden kann, wenn eine beliebige Ordnungs- oder Nachfolgerrelation zum Graphen hinzugefügt wird. Andererseits zeigen wir, dass das Problem für ordnungsinvariante monadische Logik zweiter Stufe auf allen Klassen, für die bekannt ist, dass es für monadische Logik zweiter Stufe effizient gelöst werden kann, auch effizient gelöst werden kann. Wir zeigen, dass das Problem für nachfolgerinvariante Prädikatenlogik auf planaren Graphen effizient gelöst werden kann.
Sparsity
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Author : Jaroslav Nešetřil
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-04-24
Sparsity written by Jaroslav Nešetřil and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-04-24 with Mathematics categories.
This is the first book devoted to the systematic study of sparse graphs and sparse finite structures. Although the notion of sparsity appears in various contexts and is a typical example of a hard to define notion, the authors devised an unifying classification of general classes of structures. This approach is very robust and it has many remarkable properties. For example the classification is expressible in many different ways involving most extremal combinatorial invariants. This study of sparse structures found applications in such diverse areas as algorithmic graph theory, complexity of algorithms, property testing, descriptive complexity and mathematical logic (homomorphism preservation,fixed parameter tractability and constraint satisfaction problems). It should be stressed that despite of its generality this approach leads to linear (and nearly linear) algorithms. Jaroslav Nešetřil is a professor at Charles University, Prague; Patrice Ossona de Mendez is a CNRS researcher et EHESS, Paris. This book is related to the material presented by the first author at ICM 2010.
Nowhere Dense Classes Of Graphs
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Author : Sebastian Siebertz
language : en
Publisher:
Release Date : 2016
Nowhere Dense Classes Of Graphs written by Sebastian Siebertz and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016 with categories.
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Author :
language : en
Publisher: World Scientific
Release Date :
written by and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on with categories.
Graph Theoretic Concepts In Computer Science
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Author : Ernst W. Mayr
language : en
Publisher: Springer
Release Date : 2016-08-04
Graph Theoretic Concepts In Computer Science written by Ernst W. Mayr and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-08-04 with Computers categories.
This book constitutes revised selected papers from the 41st International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2015, held in Garching, Germany, in June 2015. The 32 papers presented in this volume were carefully reviewed and selected from 79 submissions. They were organized in topical sections named: invited talks; computational complexity; design and analysis; computational geometry; structural graph theory; graph drawing; and fixed parameter tractability.
Geometry Structure And Randomness In Combinatorics
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Author : Jiří Matousek
language : en
Publisher: Springer
Release Date : 2015-04-09
Geometry Structure And Randomness In Combinatorics written by Jiří Matousek and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-04-09 with Mathematics categories.
This book collects some surveys on current trends in discrete mathematics and discrete geometry. The areas covered include: graph representations, structural graphs theory, extremal graph theory, Ramsey theory and constrained satisfaction problems.
Graph Theoretic Concepts In Computer Science
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Author : Hans L. Bodlaender
language : en
Publisher: Springer
Release Date : 2017-11-01
Graph Theoretic Concepts In Computer Science written by Hans L. Bodlaender and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-11-01 with Computers categories.
This book constitutes the revised selected papers of the 43rd International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2017, held in Eindhoven, The Netherlands, in June 2017. The 31 full papers presented in this volume were carefully reviewed and selected from 71 submissions. They cover a wide range of areas, aiming at connecting theory and applications by demonstrating how graph-theoretic concepts can be applied in various areas of computer science. Another focus is on presenting recent results and on identifying and exploring promising directions of future research.
A Unified Approach To Structural Limits And Limits Of Graphs With Bounded Tree Depth
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Author : Jaroslav Nešetřil
language : en
Publisher: American Mathematical Soc.
Release Date : 2020-04-03
A Unified Approach To Structural Limits And Limits Of Graphs With Bounded Tree Depth written by Jaroslav Nešetřil and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-04-03 with Education categories.
In this paper the authors introduce a general framework for the study of limits of relational structures and graphs in particular, which is based on a combination of model theory and (functional) analysis. The authors show how the various approaches to graph limits fit to this framework and that the authors naturally appear as “tractable cases” of a general theory. As an outcome of this, the authors provide extensions of known results. The authors believe that this puts these into a broader context. The second part of the paper is devoted to the study of sparse structures. First, the authors consider limits of structures with bounded diameter connected components and prove that in this case the convergence can be “almost” studied component-wise. They also propose the structure of limit objects for convergent sequences of sparse structures. Eventually, they consider the specific case of limits of colored rooted trees with bounded height and of graphs with bounded tree-depth, motivated by their role as “elementary bricks” these graphs play in decompositions of sparse graphs, and give an explicit construction of a limit object in this case. This limit object is a graph built on a standard probability space with the property that every first-order definable set of tuples is measurable. This is an example of the general concept of modeling the authors introduce here. Their example is also the first “intermediate class” with explicitly defined limit structures where the inverse problem has been solved.
Sofsem 2024 Theory And Practice Of Computer Science
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Author : Henning Fernau
language : en
Publisher: Springer Nature
Release Date : 2024-02-07
Sofsem 2024 Theory And Practice Of Computer Science written by Henning Fernau and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-02-07 with Computers categories.
This book constitutes the proceedings of the 49th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2024, held in Cochem, Germany, in February 2024. The 33 full papers presented in this book were carefully reviewed and selected from 81 submissions. The book also contains one invited talk in full paper length. They focus on original research and challenges in foundations of computer science including algorithms, AI-based methods, computational complexity, and formal models.
Algorithms And Data Structures
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Author : Frank Dehne
language : en
Publisher: Springer
Release Date : 2013-07-11
Algorithms And Data Structures written by Frank Dehne and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-07-11 with Computers categories.
This book constitutes the refereed proceedings of the 13th Algorithms and Data Structures Symposium, WADS 2013, held in London, ON, Canada, August 2013. The Algorithms and Data Structures Symposium - WADS (formerly "Workshop on Algorithms and Data Structures") is intended as a forum for researchers in the area of design and analysis of algorithms and data structures. The 44 revised full papers presented in this volume were carefully reviewed and selected from 139 submissions. The papers present original research on algorithms and data structures in all areas, including bioinformatics, combinatorics, computational geometry, databases, graphics, and parallel and distributed computing.