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 {\phi} with parameters we are able to define, in every graph G in C, a relation R that satisfies some hereditary first-order assertion {\psi}. Then we are able to find a first-order formula {\phi}' that has the same property, but additionally is finitary: there is finite bound k such that in every graph G in C, different choices of parameters give only at most k different relations R that can be defined using {\phi}'. 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(n^2)-time algorithm that given an n-vertex graph G in 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(n^2)-time isomorphism test and canonization algorithm for any fixed class of bounded shrubdepth.

翻译：我们利用来自稳定性理论的模型论工具推导出一个称为“有限替代引理”的结果，其直观含义如下：假设我们在一个稳定的图类C中工作，并利用带参数的一阶公式φ，能够在C中的每个图G上定义一个满足某个遗传一阶断言ψ的关系R。那么，我们可以找到一个具有相同性质但额外具有有限性的一阶公式φ'：存在一个有限界k，使得在C中的每个图G中，不同的参数选择最多只能定义出k种不同的关系R。我们运用有限替代引理推导出关于某些结构良好的图类中存在典范分解的两个推论：1. 在刻画无处稠密图类的Splitter游戏和刻画单子稳定图类的Flipper游戏中，分别存在一个可由游戏历史的一阶逻辑定义的获胜策略（对于Splitter和Flipper而言）。因此，该策略是典范的。2. 对于任何固定的一类有界灌木深度图类C，存在一个O(n²)时间的算法：给定C中一个n个顶点的图G，该算法以同构不变的方式计算出一个有界树深度的结构H，使得G可在H中解释。该结果的一个推论是：任何固定有界灌木深度图类均可在O(n²)时间内完成同构测试和典范化。