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Separability Properties of Monadically Dependent Graph Classes

Authors Édouard Bonnet , Samuel Braunfeld , Ioannis Eleftheriadis , Colin Geniet , Nikolas Mählmann , Michał Pilipczuk , Wojciech Przybyszewski , Szymon Toruńczyk

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Finite Model Theory
Keywords
  • Structural graph theory
  • Monadic dependence

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Abstract

A graph class 𝒞 is monadically dependent if one cannot interpret all graphs in colored graphs from 𝒞 using a fixed first-order interpretation. We prove that monadically dependent classes can be exactly characterized by the following property, which we call flip-separability: for every r ∈ ℕ, ε > 0, and every graph G ∈ 𝒞 equipped with a weight function on vertices, one can apply a bounded (in terms of 𝒞,r,ε) number of flips (complementations of the adjacency relation on a subset of vertices) to G so that in the resulting graph, every radius-r ball contains at most an ε-fraction of the total weight. On the way to this result, we introduce a robust toolbox for working with various notions of local separations in monadically dependent classes.

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Édouard Bonnet, Samuel Braunfeld, Ioannis Eleftheriadis, Colin Geniet, Nikolas Mählmann, Michał Pilipczuk, Wojciech Przybyszewski, and Szymon Toruńczyk. Separability Properties of Monadically Dependent Graph Classes. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 147:1-147:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.147

Author Details

Édouard Bonnet
  • CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR 5668, Lyon, France
Samuel Braunfeld
  • Computer Science Institute, Charles University, Prague, Czech Republic
  • Institute of Computer Science, The Czech Academy of Sciences, Prague, Czech Republic
Ioannis Eleftheriadis
  • Department of Computer Science and Technology, University of Cambridge, UK
Colin Geniet
  • Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea
Nikolas Mählmann
  • University of Bremen, Germany
Michał Pilipczuk
  • University of Warsaw, Poland
Wojciech Przybyszewski
  • University of Warsaw, Poland
Szymon Toruńczyk
  • University of Warsaw, Poland

Funding

ÉB and CG were supported by the ANR projects TWIN-WIDTH (ANR-21-CE48-0014-01) and DIGRAPHS (ANR-19-CE48-0013-01). SB received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 810115 - Dynasnet), and by Project 24-12591M of the Czech Science Foundation (GAČR), and supported partly by the long-term strategic development financing of the Institute of Computer Science (RVO: 67985807). IE was supported by a George and Marie Vergottis Scholarship awarded through Cambridge Trust, an Onassis Foundation Scholarship, and a Robert Sansom studentship. CG was further supported by the Institute for Basic Science (IBS-R029-C1). NM was supported by the German Research Foundation (DFG) with grant agreement No. 444419611. MP and WP were supported by the project BOBR that has received funding from ERC under the European Union’s Horizon 2020 research and innovation programme, grant agreement No. 948057. SzT received funding from ERC grant BUKA (No. 101126229).

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