Twin-width I: tractable FO model checking

Bonnet, Édouard; Kim, Eun Jung; Thomassé, Stéphan; Watrigant, Rémi

doi:10.1109/focs46700.2020.00062

Cited by 67 publications

(208 citation statements)

References 15 publications

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“…The indirect nature of the proof, which does not immediately provide an O(1)-sequence from a partition sequence with bounded oriented width, suggests that something non-trivial is at play. Indeed this greatly simplifies, as we then exemplify, the proof that K t -minor free graphs have bounded twin-width presented in [7]. We also observe that planar graphs have oriented twin-width at most 9.…”

Section: Introductionmentioning

confidence: 57%

“…The proof simply revisits the equivalence between bounded twin-width and so-called O(1)-mixed freeness (see Section 2), and can almost integrally be found in [7]. Our contribution here is mainly conceptual, in identifying the overlooked oriented twin-width.…”

Section: Introductionmentioning

confidence: 90%

“…1 Classes of bounded twin-width are surprisingly diverse. They include for instance classes of bounded treewidth, or even bounded rank-width, proper minor-closed classes, hereditary proper subclasses of permutation graphs, subgraphs of O(1)-dimensional grids [7], as well as Ω(log n)-subdivisions of n-vertex graphs, classes with Let us sketch how our algorithm goes. Instead of parsing a clique-width expression (or a tree-decomposition), we scan the contraction sequence from G = G n to G 1 .…”

Section: Introductionmentioning

confidence: 99%

“…Dynamic programming over contraction sequences tackles in one sweep problems that were seemingly as different and required as disparate techniques as k-Subgraph Isomorphism on planar graphs 3 and q-Coloring on graphs of bounded rank-width [13]. Another realization of that collapse is a similar algorithm efficiently solving FO model checking on graphs of bounded twin-width given with O(1)-sequences [7], and MSO model checking on graphs of bounded rank-width/bounded component twin-width. This motivates searching for characterizations or generalizations of classes of bounded expansion or nowhere dense classes by means of contraction sequences, as a way to push further their unifying power.…”

Section: Introductionmentioning

confidence: 99%

“…Bounded-degree and bounded-expansion are reasonably "tractable" classes with unbounded twinwidth [4]. We say that a class C is collapsible to a class D if graphs of C admit partial O(1)-sequences to (tri)graphs in D. We showcase the flexibility of the FO model-checking algorithm in [7]: Collapsible classes to bounded degree and collapsible classes to bounded expansion admit respectively a fixed-parameter tractable FO and ∃FO modelchecking algorithm, provided a corresponding partial O(1)-sequence is given. This is a relatively elementary fusion of the algorithm in [7] and classic techniques from the meta-algorithmic toolbox, namely Gaifman's locality theorem and low tree-depth covers.…”

Section: Introductionmentioning

confidence: 99%

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Twin-width VI: the lens of contraction sequences

Bonnet¹,

Kim²,

Reinald³

et al. 2022

Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

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A contraction sequence of a graph consists of iteratively merging two of its vertices until only one vertex remains. The recently introduced twin-width graph invariant is based on contraction sequences. More precisely, if one puts error edges, henceforth red edges, between two vertices representing non-homogeneous subsets, the twin-width is the minimum integer d such that a contraction sequence exists that keeps red degree at most d. By changing the condition imposed on the trigraphs (i.e., graphs with some edges being red) and possibly slightly tweaking the notion of contractions, we show how to characterize the well-established bounded rankwidth, tree-width, linear rank-width, path-width -usually defined in the framework of branch-decompositions-, and proper minor-closed classes by means of contraction sequences.Contraction sequences hold a crucial advantage over branch-decompositions: While one can scale down contraction sequences to capture classical width notions, the more general bounded twin-width goes beyond their scope, as it contains planar graphs in particular, a class with unbounded rank-width. As an application we give a transparent alternative proof of the celebrated Courcelle's theorem (actually of its generalization by Courcelle, Makowsky, and Rotics), that MSO2 (resp. MSO1) model checking on graphs with bounded tree-width (resp. bounded rank-width) is fixed-parameter tractable in the size of the input sentence. We are hopeful that our characterizations can help in other contexts.We then explore new avenues along the general theme of contraction sequences both in order to refine the landscape between bounded tree-width and bounded twin-width (via spanning twin-width) and to capture more general classes than bounded twin-width. To this end, we define an oriented version of twin-width, where appearing red edges are oriented away from the newly contracted vertex, and the mere red out-degree should remain bounded. Surprisingly, classes of bounded oriented twin-width coincide with those of bounded twin-width. This greatly simplifies the task of showing that a class has bounded twin-width. As an example, using a lemma by Norine, Seymour, Thomas, and Wollan, we give a 5-line proof that Kt-minor free graphs have bounded twin-width. Without oriented twin-width, this fact was shown by a somewhat intricate 4page proof in the first paper of the series. Finally we explore the concept of partial contraction sequences, instead of terminating on a single-vertex graph, the sequence ends when reaching a particular target class. We show that FO model checking (resp. ∃FO model checking) is fixed-parameter tractable on classes with partial contraction sequences to a class of bounded degree (resp. bounded expansion), provided such a sequence is given. Efficiently finding such partial sequences could turn out simpler than finding a (complete) sequence.

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Section: Introductionmentioning

confidence: 57%

Section: Introductionmentioning

confidence: 90%