Twin-width I: tractable FO model checking

by   Édouard Bonnet, et al.

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 ϕ evaluates to true for a given binary structure G on a domain D, is FPT in |ϕ| 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,|ϕ|) · |D| 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].


page 1

page 2

page 3

page 4


First Order Logic and Twin-Width in Tournaments and Dense Oriented Graphs

We characterise the classes of tournaments with tractable first-order mo...

Twin-Width is Linear in the Poset Width

Twin-width is a new parameter informally measuring how diverse are the n...

Twin-width III: Max Independent Set and Coloring

We recently introduced the graph invariant twin-width, and showed that f...

Flip-width: Cops and Robber on dense graphs

We define new graph parameters, called flip-width, that generalize treew...

Twin-width VI: the lens of contraction sequences

A contraction sequence of a graph consists of iteratively merging two of...

Twin-width and types

We study problems connected to first-order logic in graphs of bounded tw...

Twin-width V: linear minors, modular counting, and matrix multiplication

We continue developing the theory around the twin-width of totally order...

Please sign up or login with your details

Forgot password? Click here to reset