Twin-width IV: ordered graphs and matrices

TL;DR

This paper characterizes bounded twin-width in ordered graphs and matrices, extending key combinatorial and algorithmic results, including growth rates, fixed-parameter algorithms, and model checking classifications.

Contribution

It provides new characterizations of bounded twin-width, generalizes the Stanley-Wilf conjecture to matrix classes, and offers fixed-parameter algorithms and classifications for ordered structures.

Findings

Hereditary classes of matrices are either extremely large or exponentially bounded in size.

A fixed-parameter approximation algorithm for twin-width on ordered graphs is developed.

A full classification of fixed-parameter tractable first-order model checking on ordered classes is achieved.

Abstract

We establish a list of characterizations of bounded twin-width for hereditary, totally ordered binary structures. This has several consequences. First, it allows us to show that a (hereditary) class of matrices over a finite alphabet either contains at least $n!$ matrices of size $n \times n$, or at most $c^n$ for some constant $c$. This generalizes the celebrated Stanley-Wilf conjecture/Marcus-Tardos theorem from permutation classes to any matrix class over a finite alphabet, answers our small conjecture [SODA '21] in the case of ordered graphs, and with more work, settles a question first asked by Balogh, Bollob\'as, and Morris [Eur. J. Comb. '06] on the growth of hereditary classes of ordered graphs. Second, it gives a fixed-parameter approximation algorithm for twin-width on ordered graphs. Third, it yields a full classification of fixed-parameter tractable first-order model checking on hereditary classes of ordered binary structures. Fourth, it provides a model-theoretic characterization of classes with bounded twin-width.