Canonical Decompositions in Monadically Stable and Bounded Shrubdepth Graph Classes

Authors Pierre Ohlmann , Michał Pilipczuk , Wojciech Przybyszewski , Szymon Toruńczyk

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Author Details

Pierre Ohlmann
  • Institute of Informatics, University of Warsaw, Poland
Michał Pilipczuk
  • Institute of Informatics, University of Warsaw, Poland
Wojciech Przybyszewski
  • Institute of Informatics, University of Warsaw, Poland
Szymon Toruńczyk
  • Institute of Informatics, University of Warsaw, Poland


We are indebted to Pierre Simon for enlightening discussions about classic results in stability theory that greatly helped in deriving the results presented in this work. We also thank Jakub Gajarský, Rose McCarty, and Marek Sokołowski for their contribution in discussions around the topic of this work.

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Pierre Ohlmann, Michał Pilipczuk, Wojciech Przybyszewski, and Szymon Toruńczyk. Canonical Decompositions in Monadically Stable and Bounded Shrubdepth Graph Classes. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 135:1-135:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We use model-theoretic tools originating from stability theory to derive a result we call the Finitary Substitute Lemma, which intuitively says the following. Suppose we work in a stable graph class 𝒞, and using a first-order formula φ with parameters we are able to define, in every graph G ∈ 𝒞, a relation R that satisfies some hereditary first-order assertion ψ. Then we are able to find a first-order formula φ' that has the same property, but additionally is finitary: there is finite bound k ∈ ℕ such that in every graph G ∈ 𝒞, different choices of parameters give only at most k different relations R that can be defined using φ'. We use the Finitary Substitute Lemma to derive two corollaries about the existence of certain canonical decompositions in classes of well-structured graphs. - We prove that in the Splitter game, which characterizes nowhere dense graph classes, and in the Flipper game, which characterizes monadically stable graph classes, there is a winning strategy for Splitter, respectively Flipper, that can be defined in first-order logic from the game history. Thus, the strategy is canonical. - We show that for any fixed graph class 𝒞 of bounded shrubdepth, there is an 𝒪(n²)-time algorithm that given an n-vertex graph G ∈ 𝒞, computes in an isomorphism-invariant way a structure H of bounded treedepth in which G can be interpreted. A corollary of this result is an 𝒪(n²)-time isomorphism test and canonization algorithm for any fixed class of bounded shrubdepth.

Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • Model Theory
  • Stability Theory
  • Shrubdepth
  • Nowhere Dense
  • Monadically Stable


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