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

Authors Albert Atserias, Stephan Kreutzer, Marc Noy

Thumbnail PDF


  • Filesize: 472 kB
  • 14 pages

Document Identifiers

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

Cite AsGet BibTex

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


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. E. A. Bender, Z. Gao, L. B. Richmond, and N. Wormald. Asymptotic properties of rooted 3-connected maps on surfaces. J. Austral. Math. Soc., Series A, 60:31-41, 1996. Google Scholar
  2. G. Chapuy, E. Fusy, O. Giménez, B. Mohar, and M. Noy. Asymptotic enumeration and limit laws for graphs of fixed genus. J. Combin. Theory Ser. A, 118:748-777, 2011. Google Scholar
  3. B. Courcelle. The monadic second-order logic of graphs XI: Hierarchical decompositions of connected graphs. Theoretical Computer Science, 224:35-58, 1999. Google Scholar
  4. B. Courcelle. The monadic second-order logic of graphs xii: planar graphs and planar maps. Theoretical Computer Science (TCS), 237:1-32, 2000. Google Scholar
  5. B. Courcelle and J. Engelfriet. Graph Structure and Monadic Second-Order Logic. Cambridge University Press, 2012. Google Scholar
  6. R. Diestel. Graph Theory. Springer-Verlag, 4th edition, 2010. Google Scholar
  7. M.N. Ellingham. Spanning paths, cycles and walks for graphs on surfaces. In Congressus Numerantium, volume 115, pages 55-90, 1996. Google Scholar
  8. O. Giménez, M. Noy, and J. Rué. Graph classes with given 3-connected components: asymptotic enumeration and random graphs. Random Structures Algorithms, 42(4):438-479, 2013. Google Scholar
  9. P. Heinig, T. Müller, M. Noy, and A. Taraz. Logical limit laws for minor-closed classes of graphs. J. Combin. Theory Ser. B, to appear. Google Scholar
  10. B. Mohar and C. Thomassen. Graphs on Surfaces. Johns Hopkins University Press, 2001. Google Scholar
  11. K. Ota and K. Ozeki. Spanning trees in 3-connected K_3,t-minor-free graphs. J. Combin. Theory Ser. B, 102(5):1179-1188, 2012. Google Scholar
  12. L. B. Richmond and N. C. Wormald. Almost all maps are asymmetric. J. Combin. Theory Ser. B, 63(1):1-7, 1995. Google Scholar
  13. N. Robertson and R. Vitray. Representativity of surface embeddings. In Paths, flows, and VLSI-layout (Bonn, 1988), volume 9 of Algorithms Combin., pages 293-328. Springer, Berlin, 1990. Google Scholar