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We use model-theoretic tools originating from stability theory to derive aresult we call the Finitary Substitute Lemma, which intuitively says thefollowing. Suppose we work in a stable graph class C, and using a first-orderformula {\phi} with parameters we are able to define, in every graph G in C, arelation R that satisfies some hereditary first-order assertion {\psi}. Then weare able to find a first-order formula {\phi}' that has the same property, butadditionally is finitary: there is finite bound k such that in every graph G inC, different choices of parameters give only at most k different relations Rthat can be defined using {\phi}'. We use the Finitary Substitute Lemma toderive two corollaries about the existence of certain canonical decompositionsin classes of well-structured graphs.  - We prove that in the Splitter game, which characterizes nowhere dense graphclasses, and in the Flipper game, which characterizes monadically stable graphclasses, there is a winning strategy for Splitter, respectively Flipper, thatcan be defined in first-order logic from the game history. Thus, the strategyis canonical.  - We show that for any fixed graph class C of bounded shrubdepth, there is anO(n^2)-time algorithm that given an n-vertex graph G in C, computes in anisomorphism-invariant way a structure H of bounded treedepth in which G can beinterpreted. A corollary of this result is an O(n^2)-time isomorphism test andcanonization algorithm for any fixed class of bounded shrubdepth.

Canonical decompositions in monadically stable and bounded shrubdepth graph classes