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Descriptive Complexity, Canonisation, and Definable Graph Structure Theory

Part of Lecture Notes in Logic

  • Date Published: August 2017
  • availability: Available
  • format: Hardback
  • isbn: 9781107014527


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About the Authors
  • Descriptive complexity theory establishes a connection between the computational complexity of algorithmic problems (the computational resources required to solve the problems) and their descriptive complexity (the language resources required to describe the problems). This groundbreaking book approaches descriptive complexity from the angle of modern structural graph theory, specifically graph minor theory. It develops a 'definable structure theory' concerned with the logical definability of graph theoretic concepts such as tree decompositions and embeddings. The first part starts with an introduction to the background, from logic, complexity, and graph theory, and develops the theory up to first applications in descriptive complexity theory and graph isomorphism testing. It may serve as the basis for a graduate-level course. The second part is more advanced and mainly devoted to the proof of a single, previously unpublished theorem: properties of graphs with excluded minors are decidable in polynomial time if, and only if, they are definable in fixed-point logic with counting.

    • Contains original results which use methods from finite model theory to show the interaction between graph theory and computational complexity
    • Includes a wealth of new results, not previously published
    • Provides a reference for future research in this area
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    Reviews & endorsements

    'The book is divided evenly into two parts. Part I gives background and definitions of the main notions, and makes the book self-contained. Many results from descriptive complexity theory, and the author's earlier results, are clearly presented. Part II is devoted to the main theorem about graphs with excluded minors. The book ends with a symbol index and an index.' Pascal Michel, Mathematical Reviews

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    Product details

    • Date Published: August 2017
    • format: Hardback
    • isbn: 9781107014527
    • length: 554 pages
    • dimensions: 235 x 160 x 36 mm
    • weight: 0.88kg
    • contains: 60 b/w illus.
    • availability: Available
  • Table of Contents

    1. Introduction
    Part I. The Basic Theory:
    2. Background from graph theory and logic
    3. Descriptive complexity
    4. Treelike decompositions
    5. Definable decompositions
    6. Graphs of bounded tree width
    7. Ordered treelike decompositions
    8. 3-Connected components
    9. Graphs embeddable in a surface
    Part II. Definable Decompositions of Graphs with Excluded Minors:
    10. Quasi-4-connected components
    11. K5-minor free graphs
    12. Completions of pre-decompositions
    13. Almost planar graphs
    14. Almost planar completions
    15. Almost embeddable graphs
    16. Decompositions of almost embeddable graphs
    17. Graphs with excluded minors
    18. Bits and pieces
    Appendix. Robertson and Seymour's version of the local structure theorem
    Symbol index

  • Author

    Martin Grohe, RWTH Aachen University, Germany
    Martin Grohe is a Professor of Theoretical Computer Science at RTWH Aachen University, Germany, where he holds the Chair for Logic and the Theory of Discrete Systems. His research interests are in theoretical computer science interpreted broadly, including logic, algorithms and complexity, graph theory, and database theory.

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