License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2018.116
URN: urn:nbn:de:0030-drops-91206
URL: https://drops.dagstuhl.de/opus/volltexte/2018/9120/
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Atserias, Albert ; Kreutzer, Stephan ; Noy, Marc

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

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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.

BibTeX - Entry

@InProceedings{atserias_et_al:LIPIcs:2018:9120,
  author =	{Albert Atserias and Stephan Kreutzer and Marc Noy},
  title =	{{On Zero-One and Convergence Laws for Graphs Embeddable on a Fixed Surface}},
  booktitle =	{45th International Colloquium on Automata, Languages, and  Programming (ICALP 2018)},
  pages =	{116:1--116:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9120},
  URN =		{urn:nbn:de:0030-drops-91206},
  doi =		{10.4230/LIPIcs.ICALP.2018.116},
  annote =	{Keywords: topological graph theory, monadic second-order logic, random graphs, zero-one law, convergence law}
}

Keywords: topological graph theory, monadic second-order logic, random graphs, zero-one law, convergence law
Collection: 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)
Issue Date: 2018
Date of publication: 04.07.2018


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