| Description |
1 online resource (viii, 270 pages) : illustrations |
| Contents |
Intro -- Preface -- Contents -- 1 Beyond Planar Graphs: Introduction -- 1.1 Beyond Planar Graphs: Edge Density -- 1.2 Computational Complexity: NP-Hardness -- 1.3 Polynomial-Time Testing Algorithm -- 1.4 Straight-Line Drawing -- 1.5 Outlook and Open Problem -- References -- 2 Quantitative Restrictions on Crossing Patterns -- 2.1 Introduction -- 2.2 Simple Graphs and Simple Topological Graphs -- 2.2.1 Simple Graphs Versus Multigraphs -- 2.2.2 t-Simple Topological Graphs -- 2.3 Overview of Graph Classes -- 2.4 Density -- 2.5 Inclusions -- 2.6 Recognition Algorithms -- 2.7 Open Problems |
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4.7 Difference from Optimal 1-Planar Graphs -- 4.8 Open Problems -- References -- 5 Algorithms for 1-Planar Graphs -- 5.1 Introduction -- 5.2 Testing Maximal 1-Planarity -- 5.3 Testing Outer-1-Planarity -- 5.4 Straight-Line Drawing Algorithm for 1-Planar Graphs -- 5.5 Recent Progress -- References -- 6 Edge Partitions and Visibility Representations of 1-planar Graphs -- 6.1 Introduction -- 6.2 Basic Definitions and Notation -- 6.3 Recognition and Structural Properties of 1-planar Graphs -- 6.4 Edge Partitions of 1-plane Graphs -- 6.4.1 Summary of Results -- 6.4.2 A Glimpse of a Proof Technique |
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6.5 Visibility Representations of 1-plane Graphs -- 6.5.1 Rectangle and Ortho-Polygon Visibility Representations -- 6.5.2 A Glimpse of a Proof Technique -- 6.5.3 Further Results on Visibility Representations of 1-plane Graphs -- 6.6 Concluding Remarks and Open Problems -- References -- 7 k-Planar Graphs -- 7.1 Introduction -- 7.2 Examples of k-Planar Graphs with High Density -- 7.3 Density Results and Their Implications to the Crossing Lemma -- 7.3.1 Two Important Implications -- 7.4 Interesting Subclasses -- 7.5 Relationship with k-quasi-planarity -- References -- 8 Fan-Planar Graphs |
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8.1 Introduction -- 8.2 Examples of Dense Fan-Planar Graphs -- 8.3 Relationships with Other Classes of Beyond-Planar Graphs -- 8.4 Density Results -- 8.5 NP-Hardness Results -- 8.6 Algorithmic Results -- 8.7 Open Problems -- References -- 9 Right Angle Crossing Drawings of Graphs -- 9.1 Introduction -- 9.2 Basic Terminology and Properties -- 9.3 Edge Density -- 9.4 Testing and Drawing Algorithms -- 9.4.1 RAC Drawings with Bends -- 9.4.2 Straight-Line RAC Drawings -- 9.4.3 RAC Drawings of Planar Graphs -- 9.5 Inclusion Relationships with 1-Planar Graphs -- 9.6 Concluding Remarks -- References |
| Summary |
This book is the first general and extensive review on the algorithmics and mathematical results of beyond planar graphs. Most real-world data sets are relational and can be modelled as graphs consisting of vertices and edges. Planar graphs are fundamental for both graph theory and graph algorithms and are extensively studied. Structural properties and fundamental algorithms for planar graphs have been discovered. However, most real-world graphs, such as social networks and biological networks, are non-planar. To analyze and visualize such real-world networks, it is necessary to solve fundamental mathematical and algorithmic research questions on sparse non-planar graphs, called beyond planar graphs.This book is based on the National Institute of Informatics (NII) Shonan Meeting on algorithmics on beyond planar graphs held in Japan in November, 2016. The book consists of 13 chapters that represent recent advances in various areas of beyond planar graph research. The main aims and objectives of this book include 1) to timely provide a state-of-the-art survey and a bibliography on beyond planar graphs; 2) to set the research agenda on beyond planar graphs by identifying fundamental research questions and new research directions; and 3) to foster cross-disciplinary research collaboration between computer science (graph drawing and computational geometry) and mathematics (graph theory and combinatorics). New algorithms for beyond planar graphs will be in high demand by practitioners in various application domains to solve complex visualization problems. This book therefore will be a valuable resource for researchers in graph theory, algorithms, and theoretical computer science, and will stimulate further deep scientific investigations into many areas of beyond planar graphs |
| Notes |
Contains a selection of book chapters initiated from the Shonan Workshop No-089 on Beyond Planar Graphs, held as part of the The NII (National Institute of Informatics) Shonan Meeting, November 27–December 1, 2016 in Shonan, Japan |
| Bibliography |
Includes bibliographical references and index |
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References -- 3 Quasi-planar Graphs -- 3.1 Introduction -- 3.2 The Size of k-Quasi-planar Graphs -- 3.2.1 3- and 4-Quasi-planar Graphs -- 3.2.2 k-Quasi-planar Graphs for k ge5 -- 3.2.3 Restricted Drawings -- 3.2.4 Lower Bounds -- 3.3 Relationships with Other Classes of Graphs -- 3.3.1 Beyond-Planar Graphs -- 3.3.2 Complete Graphs -- 3.4 Computational Aspects of k-Quasi-planar Graphs -- References -- 4 1-Planar Graphs -- 4.1 Definition and Basic Results -- 4.2 Connectivity -- 4.3 Planarization -- 4.4 Edge Density -- 4.5 Minors and Subgraphs -- 4.6 Re-embeddings of 1-Planar Graphs |
| Subject |
Graph algorithms.
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Graph theory -- Data processing.
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Graph algorithms
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Graph theory -- Data processing
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| Form |
Electronic book
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| Author |
Hong, Seok-Hee, editor.
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Tokuyama, Takeshi, editor.
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| ISBN |
9789811565335 |
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9811565333 |
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