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Indiscernibles and Flatness in Monadically Stable and Monadically NIP Classes

Authors Jan Dreier , Nikolas Mählmann , Sebastian Siebertz , Szymon Toruńczyk

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Author Details

Jan Dreier
  • TU Wien, Austria
Nikolas Mählmann
  • Universität Bremen, Germany
Sebastian Siebertz
  • Universität Bremen, Germany
Szymon Toruńczyk
  • University of Warsaw, Poland

Funding

  • Mählmann, Nikolas: supported by the German Research Foundation (DFG) with grant greement No. 444419611.
  • Toruńczyk, Szymon: supported by the project BOBR that is funded from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme with grant agreement No. 948057.

Acknowledgements

We thank Édouard Bonnet, Jakub Gajarský, Stephan Kreutzer, Amer E. Mouawad and Alexandre Vigny for their valuable contributions to this paper. In particular, we thank Jakub Gajarský and Stephan Kreutzer for suggesting the notion of flip-flatness and providing a proof for the cases r = 1,2.

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Jan Dreier, Nikolas Mählmann, Sebastian Siebertz, and Szymon Toruńczyk. Indiscernibles and Flatness in Monadically Stable and Monadically NIP Classes. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 125:1-125:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.125

Abstract

Monadically stable and monadically NIP classes of structures were initially studied in the context of model theory and defined in logical terms. They have recently attracted attention in the area of structural graph theory, as they generalize notions such as nowhere denseness, bounded cliquewidth, and bounded twinwidth.
Our main result is the - to the best of our knowledge first - purely combinatorial characterization of monadically stable classes of graphs, in terms of a property dubbed flip-flatness. A class C of graphs is flip-flat if for every fixed radius r, every sufficiently large set of vertices of a graph G ∈ C contains a large subset of vertices with mutual distance larger than r, where the distance is measured in some graph G' that can be obtained from G by performing a bounded number of flips that swap edges and non-edges within a subset of vertices. Flip-flatness generalizes the notion of uniform quasi-wideness, which characterizes nowhere dense classes and had a key impact on the combinatorial and algorithmic treatment of nowhere dense classes. To obtain this result, we develop tools that also apply to the more general monadically NIP classes, based on the notion of indiscernible sequences from model theory. We show that in monadically stable and monadically NIP classes indiscernible sequences impose a strong combinatorial structure on their definable neighborhoods. All our proofs are constructive and yield efficient algorithms.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Finite Model Theory
Keywords
  • stability
  • NIP
  • combinatorial characterization
  • first-order model checking

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