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On Zero-One and Convergence Laws for Graphs Embeddable on a Fixed Surface

Authors Albert Atserias, Stephan Kreutzer, Marc Noy

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graphs and surfaces
  • Mathematics of computing → Enumeration
  • Theory of computation → Finite Model Theory
Keywords
  • topological graph theory
  • monadic second-order logic
  • random graphs
  • zero-one law
  • convergence law

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Abstract

We show that for no surface except for the plane does monadic second-order logic (MSO) have a zero-one-law - and not even a convergence law - on the class of (connected) graphs embeddable on the surface. In addition we show that every rational in [0,1] is the limiting probability of some MSO formula. This strongly refutes a conjecture by Heinig et al. (2014) who proved a convergence law for planar graphs, and a zero-one law for connected planar graphs, and also identified the so-called gaps of [0,1]: the subintervals that are not limiting probabilities of any MSO formula. The proof relies on a combination of methods from structural graph theory, especially large face-width embeddings of graphs on surfaces, analytic combinatorics, and finite model theory, and several parts of the proof may be of independent interest. In particular, we identify precisely the properties that make the zero-one law work on planar graphs but fail for every other surface.

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Albert Atserias, Stephan Kreutzer, and Marc Noy. On Zero-One and Convergence Laws for Graphs Embeddable on a Fixed Surface. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 116:1-116:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ICALP.2018.116

Author Details

Albert Atserias
  • Universitat Politècnica de Catalunya, Barcelona, atserias@cs.upc.edu
Stephan Kreutzer
  • Technical University Berlin, stephan.kreutzer@tu-berlin.de
Marc Noy
  • Universitat Politècnica de Catalunya, Barcelona, marc.noy@upc.edu

Funding

  • Atserias, Albert: Partially funded by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant agreement ERC-2014-CoG 648276 (AUTAR) and MICCIN grant TIN2016-76573-C2-1P (TASSAT3).
  • Kreutzer, Stephan: Research supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC Consolidator Grant DISTRUCT, grant agreement No 648527).
  • Noy, Marc: Partially supported by Grants MTM2014-54745-P and MDM-2014-0445.

References

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