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Vertex partitioning problems on partial k-trees

  • Arvind Gupta
  • Damon Kaller
  • Sanjeev Mahajan
  • Tom Shermer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1097)

Abstract

We describe a general approach to obtain polynomial-time algorithms over partial k-trees for graph problems in which the vertex set is to be partitioned in some way. We encode these problems with formulae of the Extended Monadic Second-order (or EMS) logic. Such a formula can be translated into a polynomial-time algorithm automatically. We focus on the problem of partitioning a partial k-tree into induced subgraphs isomorphic to a fixed pattern graph; a distinct algorithm is derived for each pattern graph and each value of k. We use a “pumping lemma” to show that (for some pattern graphs) this problem cannot be encoded in the “ordinary” Monadic Second-order logic—from which a linear-time algorithm over partial k-trees would be obtained. Hence, an EMS formula is in some sense the strongest possible. As a further application of our general approach, we derive a polynomial-time algorithm to determine the maximum number of co-dominating sets into which the vertices of a partial k-tree can be partitioned. (A co-dominating set of a graph is a dominating set of its complement graph).

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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Arvind Gupta
    • 1
  • Damon Kaller
    • 1
  • Sanjeev Mahajan
    • 2
  • Tom Shermer
    • 1
  1. 1.School of Computing ScienceSimon Fraser UniversityBurnabyCanada
  2. 2.Max Planck Institut für InformatikSaarbrückenGermany

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