Twin-width I: tractable FO model checking

Abstract

Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA '14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, $K_{t^{-}}$ free unit $d$ -dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of $d$ -contractions, witness that the twin-width is at most $d$ . We show that FO model checking, that is deciding if a given first-order formula $\phi$ evaluates to true for a given binary structure $G$ on a domain $D$ , is FPT in $\vert \phi\vert$ on classes of bounded twin-width, provided the witness is given. More precisely, being given a $d$ -contraction sequence for $G$ , our algorithm runs in time $f(d,\vert \phi\vert)\cdot\vert D\vert$ where $f$ is a computable but non-elementary function. We also prove that bounded twin-width is preserved by FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarský et al. [FOCS '15].