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Graphs of Bounded Treewidth Can Be Canonized in \(\mbox{{\sf AC}$^1$}\)

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Computer Science – Theory and Applications (CSR 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6651))

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Abstract

In recent results the complexity of isomorphism testing on graphs of bounded treewidth is improved to \(\mbox{{\sf TC}$^1$}\) [17] and further to \(\mbox{{\sf LogCFL}}\) [11]. The computation of canonical forms or a canonical labeling provides more information than isomorphism testing. Whether canonization is in \(\mbox{{\sf NC}}\) or even \(\mbox{{\sf TC}$^1$}\) was stated as an open question in [18]. Köbler and Verbitsky [20] give a \(\mbox{{\sf TC}$^2$}\) canonical labeling algorithm. We show that a canonical labeling can be computed in \(\mbox{{\sf AC}$^1$}\). This is based on several ideas, e.g. that approximate tree decompositions of logarithmic depth can be obtained in logspace [15], and techniques of Lindells tree canonization algorithm [22]. We define recursively what we call a minimal description which gives with respect to some parameters in a logarithmic number of levels a canonical invariant together with an arrangement of all vertices. From this we compute a canonical labeling.

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Authors and Affiliations

  1. Inst. für Theoretische Informatik, Universität Ulm, Germany

    Fabian Wagner

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  1. Fabian Wagner
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Editors and Affiliations

  1. Steklov Institute of Mathematics at St. Petersburg, 27 Fontanka, 191023, St.Petersburg, Russia

    Alexander Kulikov

  2. Department of Mathematical Logic and Theory of Algorithms, Moscow State University, Leninskie gory, 119991, Moscow, Russia

    Nikolay Vereshchagin

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Wagner, F. (2011). Graphs of Bounded Treewidth Can Be Canonized in \(\mbox{{\sf AC}$^1$}\) . In: Kulikov, A., Vereshchagin, N. (eds) Computer Science – Theory and Applications. CSR 2011. Lecture Notes in Computer Science, vol 6651. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20712-9_16

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  • DOI: https://doi.org/10.1007/978-3-642-20712-9_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-20711-2

  • Online ISBN: 978-3-642-20712-9

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