The notions of bounded-size and quasibounded-size decompositions with bounded treedepth base classes are central to the structural theory of graph sparsity introduced by two of the authors years ago, and provide a characterization of both classes with bounded expansions and nowhere dense classes. In this paper, we first prove that the model theoretic notions of dependence and stability are, for hereditary classes of graphs, compatible with quasibounded-size decompositions, in the following sense: every hereditary class with quasibounded-size decompositions with dependent (resp.\ stable) base classes is itself dependent (resp.\ stable). This result is obtained in a more general study of ``decomposition horizons'', which are class properties compatible with quasibounded-size decompositions. We deduce that hereditary classes with quasibounded-size decompositions with bounded shrubdepth base classes are stable. In the second part of the paper, we prove the converse. Thus, we characterize stable hereditary classes of graphs as those hereditary classes that admit quasibounded-size decompositions with bounded shrubdepth base classes. This result is obtained by proving that every hereditary stable class of graphs admits almost nowhere dense quasi-bush representations, thus answering positively a conjecture of Dreier et al. These results have several consequences. For example, we show that every graph $G$ in a stable, hereditary class of graphs $\mathscr C$ has a clique or a stable set of size $\Omega_{\mathscr C,\epsilon}(|G|^{1/2-\epsilon})$, for every $\epsilon>0$, which is tight in the sense that it cannot be improved to $\Omega_{\mathscr C}(|G|^{1/2})$.

翻译：有界大小和拟有界大小（具有有界树深度基类）的分解概念是多年前由两位作者引入的图稀疏结构理论的核心，并为有界展开类和无处稠密类提供了刻画。在本文中，我们首先证明，对于图的遗传类，模型论中的依赖性和稳定性概念与拟有界大小分解是兼容的，具体而言：每个具有依赖（分别为稳定）基类的拟有界大小分解的遗传类本身也是依赖（分别为稳定）的。这一结果源于对“分解水平”的更一般研究，这些分解水平是与拟有界大小分解兼容的类性质。我们推断出，具有有界灌木深度基类的拟有界大小分解的遗传类是稳定的。在论文的第二部分，我们证明了其逆命题。因此，我们刻画了稳定的遗传类图，即那些允许具有有界灌木深度基类的拟有界大小分解的遗传类。这一结果是通过证明每个稳定遗传类图都存在几乎无处稠密的拟丛林表示获得的，从而肯定地回答了Dreier等人的一个猜想。这些结果具有若干推论。例如，我们证明，对于每个$\epsilon>0$，稳定遗传类图$\mathscr C$中的每个图$G$都具有一个团或稳定集，其大小为$\Omega_{\mathscr C,\epsilon}(|G|^{1/2-\epsilon})$，且在意义上是紧的，即不能改进为$\Omega_{\mathscr C}(|G|^{1/2})$。