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.
@InProceedings{ohlmann_et_al:LIPIcs.ICALP.2023.135, author = {Ohlmann, Pierre and Pilipczuk, Micha{\l} and Przybyszewski, Wojciech and Toru\'{n}czyk, Szymon}, title = {{Canonical Decompositions in Monadically Stable and Bounded Shrubdepth Graph Classes}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {135:1--135:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.135}, URN = {urn:nbn:de:0030-drops-181874}, doi = {10.4230/LIPIcs.ICALP.2023.135}, annote = {Keywords: Model Theory, Stability Theory, Shrubdepth, Nowhere Dense, Monadically Stable} }
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