Decomposition horizons and a characterization of stable hereditary classes of graphs

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. Strong connections of this theory with model theory led to considering first-order transductions, which are logically defined graph transformations, and to initiate a comparative study of combinatorial and model theoretical properties of graph classes, with an emphasis on the model theoretical notions of dependence (or NIP) and stability. In this paper, we first prove that 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 $\Omega_{\mathscr C}(|G|^{1/2})$.