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Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width

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Graph-Theoretic Concepts in Computer Science (WG 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1517))

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Abstract

Graphs of clique-width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we show that the (q,q-4) graphs are of clique width at most q and P 4-tidy graphs are of clique-width at most 4. Furthermore, the k-expression (for k=4 or k=q) associated with such a graph can be found in linear time.

q,q-4) graphs were introduced by Babel and Olariu (1995) and extends the class of P 4-sparse graphs. P 4-sparse graphs were introduced by Hoàng (1985) and are widely studied because of their applications in areas such as scheduling, clustering and computational semantics. Another extension of P 4-sparse graphs are the P 4-tidy graphs which were introduced by Rusu (1995).

Furthermore, we show that the class of LinEMSOL(τ 1,L) optimization problems is solvable in O(f(|V|,|E|)) time on a class of graphs of clique-width at most k in which for every graph G an expression defining it can be constructed in O(f(|V|,|E|)) time. By the above this applies in particular to (q,q – 4) graphs, P 4-tidy graphs and P 4-sparse graphs with f linear.

Finally, we show that the above results cannot be extended to MSOL(τ 2) decision and optimization problems on the vocabulary τ 2 which allow edges to be considered as elements of the domains of the graphs in question, and by that, allow quantifying over edges in addition to quantifying over vertices.

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Author information

Authors and Affiliations

  1. Laboratoire d’Informatique, Université Bordeaux-I, 33405, Talence, France

    B. Courcelle

  2. Department of Computer Science, Technion, Israel Institute of Technology, 32000, Haifa, Israel

    J. A. Makowsky & U. Rotics

Authors
  1. B. Courcelle
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  2. J. A. Makowsky
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  3. U. Rotics
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Editor information

Editors and Affiliations

  1. Department of Computer Science, ETH Zurich, ETH Zentrum, CH-8022, Zurich, Switzerland

    Juraj Hromkovič

  2. Department of Computer Science, University of Loughborough, Loughborough, UK

    Ondrej Sýkora

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© 1998 Springer-Verlag Berlin Heidelberg

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Courcelle, B., Makowsky, J.A., Rotics, U. (1998). Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width. In: Hromkovič, J., Sýkora, O. (eds) Graph-Theoretic Concepts in Computer Science. WG 1998. Lecture Notes in Computer Science, vol 1517. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10692760_1

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  • DOI: https://doi.org/10.1007/10692760_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-65195-6

  • Online ISBN: 978-3-540-49494-2

  • eBook Packages: Springer Book Archive

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