Ordered graphs of bounded twin-width

TL;DR

This paper characterizes hereditary classes of ordered graphs with bounded twin-width, linking it to model-checking tractability, NIP property, and combinatorial growth, and provides a grid theorem for such classes.

Contribution

It offers multiple equivalent characterizations of bounded twin-width in ordered graphs and establishes a grid theorem, connecting twin-width with model theory and combinatorics.

Findings

Bounded twin-width is equivalent to NIP and smallness in hereditary ordered graph classes.

Model-checking for first-order logic is fixed-parameter tractable on classes with bounded twin-width.

A gap in the growth of hereditary classes of ordered graphs is identified.

Abstract

We consider hereditary classes of graphs equipped with a total order. We provide multiple equivalent characterisations of those classes which have bounded twin-width. In particular, we prove a grid theorem for classes of ordered graphs which have unbounded twin-width. From this we derive that the model-checking problem for first-order logic is fixed-parameter tractable over a hereditary class of ordered graphs if, and -- under common complexity-theoretic assumptions -- only if the class has bounded twin-width. For hereditary classes of ordered graphs, we show that bounded twin-width is equivalent to the NIP property from model theory, as well as the smallness condition from enumerative combinatorics. We prove the existence of a gap in the growth of hereditary classes of ordered graphs. Furthermore, we provide a grid theorem which applies to all monadically NIP classes of structures (ordered or unordered), or equivalently, classes which do not transduce the class of all finite graphs.