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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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Author Details

Albert Atserias
  • Universitat Politècnica de Catalunya, Barcelona,
Stephan Kreutzer
  • Technical University Berlin,
Marc Noy
  • Universitat Politècnica de Catalunya, Barcelona,

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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)


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.

Subject Classification

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


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