Decomposition horizons and a characterization of stable hereditary classes of graphs

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985807%3A_____%2F25%3A00637116" target="_blank" >RIV/67985807:_____/25:00637116 - isvavai.cz</a>

Alternative codes found

RIV/00216208:11320/25:10511784

Result on the web

<a href="https://doi.org/10.1016/j.ejc.2025.104130" target="_blank" >https://doi.org/10.1016/j.ejc.2025.104130</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.ejc.2025.104130" target="_blank" >10.1016/j.ejc.2025.104130</a>

Result language

angličtina

Original language name

Decomposition horizons and a characterization of stable hereditary classes of graphs

Original language description

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 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 C has a clique or a stable set of size ΩC,ɛ(|G|1/2−ɛ), for every ɛ>0, which is tight in the sense that it cannot be improved to ΩC(|G|1/2).

Czech name

—

Czech description

—

Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GA21-10775S" target="_blank" >GA21-10775S: Ramsey theory in the context of group theory, model theory and topological dynamics</a><br>

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2025

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

European Journal of Combinatorics

ISSN

0195-6698

e-ISSN

1095-9971

Volume of the periodical

129

Issue of the periodical within the volume

October 2025

Country of publishing house

NL - THE KINGDOM OF THE NETHERLANDS

Number of pages

27

Pages from-to

104130

UT code for WoS article

001536475700001

EID of the result in the Scopus database

2-s2.0-85217027099