Decomposition horizons and a characterization of stable hereditary classes of graphs

TL;DR

This work introduces and studies the decomposition-horizon framework, connecting graph sparsity with model-theoretic stability. It proves that stable hereditary graph classes are exactly those admitting quasibounded-size decompositions whose base classes have bounded shrubdepth, and that dependent and stable properties are decomposition horizons; conversely, stable classes admit almost nowhere dense quasi-bush representations, resolving a conjecture of Dreier et al. The results unify structural sparsity concepts (bounded shrubdepth, SC-depth, shrubdepth-based representations) with model-theoretic notions (stability, dependence, transductions), yielding concrete consequences such as an Erdős-Hajnal property with clique or stable set size $Ω_{\mathscr C,oldsymbol{ md}}(|G|^{1/2-oldsymbol{ md}})$ for every $oldsymbol{ md}>0$. Collectively, the paper provides a general, operable framework for analyzing stable graph classes via decomposition horizons, quasi-bush representations, and transductions, with implications for algorithmic tractability and deeper structural dichotomies.

Abstract

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 $Ω_{\mathscr C,ε}(|G|^{1/2-ε})$, for every $ε>0$, which is tight in the sense that it cannot be improved to $Ω_{\mathscr C}(|G|^{1/2})$.

Figures (2)

• Figure 1: A partition of a subset $A$ of vertices of a graph $G$ with parameter $r$, as obtained by \ref{['thm:incg']}. To each part $A_{i,j}$ is associated a vertex $z_{i,j}$ (possibly not in $A$). The distance between $z_{i,j}$ and any vertex in $A_{i,j}$ is at most $r+1$, while the distance between any vertex in $A_{i,j}$ and any vertex in $A_{i,j'}$ (for $j'\neq j$) is at least $2r+3$. Consequently, $A_{i,j}=N_G^{r+1}[z_{i,j}]\cap A_i$.
• Figure 3: Construction of the graphs $H_c$ and $G_c$ (in the case $X^{p,q}$ is $C_5$).

Theorems & Definitions (64)

• Theorem 1: Taxi_stoc06POMNI
• Theorem 2: nevsetvril2011nowhere
• Theorem 3: adler2014interpreting
• Theorem 4: braunfeld2022existential
• Theorem 5: SBE_drops
• Conjecture 6
• Theorem 7: dreier2022treelike
• Conjecture 8: nevsetril2020rankwidth
• Conjecture 9
• Theorem 10