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Fixed-Point Definability and Polynomial Time on Chordal Graphs and Line Graphs

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Book cover Fields of Logic and Computation

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 6300))

Abstract

The question of whether there is a logic that captures polynomial time was formulated by Yuri Gurevich in 1988. It is still wide open and regarded as one of the main open problems in finite model theory and database theory. Partial results have been obtained for specific classes of structures. In particular, it is known that fixed-point logic with counting captures polynomial time on all classes of graphs with excluded minors. The introductory part of this paper is a short survey of the state-of-the-art in the quest for a logic capturing polynomial time.

The main part of the paper is concerned with classes of graphs defined by excluding induced subgraphs. Two of the most fundamental such classes are the class of chordal graphs and the class of line graphs. We prove that capturing polynomial time on either of these classes is as hard as capturing it on the class of all graphs. In particular, this implies that fixed-point logic with counting does not capture polynomial time on these classes. Then we prove that fixed-point logic with counting does capture polynomial time on the class of all graphs that are both chordal and line graphs.

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Authors and Affiliations

  1. Humboldt Universität zu Berlin, Germany

    Martin Grohe

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  1. Martin Grohe
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Editors and Affiliations

  1. Mathematics Department, University of Michigan, 48109-1043, Ann Arbor, MI, USA

    Andreas Blass

  2. School of Computer Science, Tel Aviv University, Ramat Aviv, 69978, Tel Aviv, Israel

    Nachum Dershowitz

  3. Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Wolfgang Reisig

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Grohe, M. (2010). Fixed-Point Definability and Polynomial Time on Chordal Graphs and Line Graphs. In: Blass, A., Dershowitz, N., Reisig, W. (eds) Fields of Logic and Computation. Lecture Notes in Computer Science, vol 6300. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15025-8_17

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  • DOI: https://doi.org/10.1007/978-3-642-15025-8_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15024-1

  • Online ISBN: 978-3-642-15025-8

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