Elsevier

Journal of Algorithms

Volume 12, Issue 2, June 1991, Pages 308-340
Journal of Algorithms

Easy problems for tree-decomposable graphs

https://doi.org/10.1016/0196-6774(91)90006-KGet rights and content

Abstract

Using a variation of the interpretability concept we show that all graph properties definable in monadic second-order logic (MS properties) with quantification over vertex and edge sets can be decided in linear time for classes of graphs of fixed bounded treewidth given a tree-decomposition. This gives an alternative proof of a recent result by Courcelle. We allow graphs with directed and/or undirected edges, labeled on edges and/or vertices with labels taken from a finite set. We extend MS properties to extended monadic second-order (EMS) problems involving counting or summing evaluations over sets definable in monadic second-order logic. Our technique allows us also to solve some EMS problems in linear time or in polynomial or pseudopolynomial time for classes of graphs of fixed bounded treewidth. Moreover, it is shown that each EMS problem is in NC for graphs of bounded treewidth. Most problems for which linear time algorithms for graphs of bounded treewidth were previously known to exist, and many others, are EMS problems.

References (45)

  • S Arnborg et al.

    Complexit of finding embeddings in a k-tree

    SIAM J. Algebra Discrete Methods

    (1987)
  • S Arnborg et al.

    Characterization and recognition of partial 3-trees

    SIAM J. Algebra Discrete Methods

    (1986)
  • U Bertele et al.

    Nonserial Dynamic Programming

    (1972)
  • H.L Bodlaender

    Dynamic Programming on Graphs with Bounded Tree-width

  • H.L Bodlaender

    Classes of Graphs with Bounded Tree-width

    (1986)
  • H.L Bodlaender

    NC-Algorithms for Graphs with Bounded Tree-width

    (1988)
  • H.L Bodlaender

    Planar Graphs with Bounded Treewidth

    (1988)
  • H.L Bodlaender

    Improved Self-reduction Algorithms from Graphs with Bounded Treewidth

    (1988)
  • R.B Borie et al.

    Automatic generation of linear algorithms from predicate calculus descriptions of problems on recursively constructed graph families

    (July 21, 1988)
  • R.B Borie

    Recursively Constructed Graph Families: Membership and Linear Algorithms

  • K Compton et al.

    A new method for proving lower bounds on the computational complexity of first-order theories, manuscript

    (1987)
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    Research supported by the Swedish Natural Sciences Research Council and the Swedish Board for Technical Development.

    Present address: FB 11/Computer Science, Univerity Dulsburg, Pf 10 15 03, C-W-4100, Dulsburg 1, Germany.

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