Stable graphs of bounded twin-width

Gajarský, Jakub; Pilipczuk, Michał; Toruńczyk, Szymon

doi:10.48550/arxiv.2107.03711

Cited by 6 publications

(17 citation statements)

References 14 publications

(28 reference statements)

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“…for cubic graphs) [BGK + 21]. Therefore our result generalizes the earlier result of [HWZ21], and essentially follows from using the structural results of [GKN + 20] instead of [GPT21].…”

Section: Small-distance Sketchingsupporting

confidence: 74%

“…For r = 1, this coincides with adjacency labelling. The natural generalization of constant-size adjacency sketches is to ask for small-distance sketches whose size depends only on r; it was shown in [Har20] that such sketches exist for trees, and in [HWZ21] that they exist for any Cartesian product graphs and any stable 3 class of bounded twin-width (including, for example, planar graphs or any proper minor-closed class; see [GPT21]).…”

Section: Motivation and Prior Workmentioning

confidence: 99%

“…We will actually show that any first-order transduction of a class with bounded expansion is first-order sketchable. It was proved in [HWZ21], using structural results of [GPT21], that any stable class of bounded twin-width is first-order sketchable. A stable class has bounded twin-width if and only if it is a transduction of a class of bounded sparse twin-width [GPT21].…”

Section: Small-distance Sketchingmentioning

confidence: 99%

“…It was proved in [HWZ21], using structural results of [GPT21], that any stable class of bounded twin-width is first-order sketchable. A stable class has bounded twin-width if and only if it is a transduction of a class of bounded sparse twin-width [GPT21]. Every class of bounded sparse twin-width has bounded expansion, but the converse does not hold (e.g.…”

Section: Small-distance Sketchingmentioning

confidence: 99%

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Sketching Distances in Monotone Graph Classes

Harms¹,

Kupavskii²

2022

Preprint

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We study the problems of adjacency sketching, small-distance sketching, and approximate distance threshold sketching for monotone classes of graphs. The problem is to obtain randomized sketches of the vertices of any graph G in the class, so that adjacency, exact distance thresholds, or approximate distance thresholds of two vertices u, v can be decided (with high probability) from the sketches of u and v, by a decoder that does not know the graph. The goal is to determine when sketches of constant size exist.We show that, for monotone classes of graphs, there is a strict hierarchy: approximate distance threshold sketches imply small-distance sketches, which imply adjacency sketches, whereas the reverse implications are each false. The existence of an adjacency sketch is equivalent to the condition of bounded arboricity, while the existence of small-distance sketches is equivalent to the condition of bounded expansion. Classes of constant expansion admit approximate distance threshold sketches, while a monotone graph class can have arbitrarily small non-constant expansion without admitting an approximate distance threshold sketch. * Partially supported by the French ANR Projects GATO (ANR-16-CE40-0009-01), GrR (ANR-18-CE40-0032), TWIN-WIDTH (ANR-21-CE48-0014-01) and by LabEx PERSYVAL-lab (ANR-11-LABX-0025).† This work was partly funded by an NSERC Canada Graduate Scholarship.

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“…for cubic graphs) [BGK + 21]. Therefore our result generalizes the earlier result of [HWZ21], and essentially follows from using the structural results of [GKN + 20] instead of [GPT21].…”

Section: Small-distance Sketchingsupporting

confidence: 74%

Section: Motivation and Prior Workmentioning

confidence: 99%

Section: Small-distance Sketchingmentioning

confidence: 99%

Section: Small-distance Sketchingmentioning

confidence: 99%

See 2 more Smart Citations

Sketching Distances in Monotone Graph Classes

Harms¹,

Kupavskii²

2022

Preprint

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“…However, deciding if the twin-width of a graph is at most four is NP-complete [6]. Bounds on twin-width are known for many graph classes [4,7,9,13,17,24,41]. In particular, twin-width is bounded on some classes of dense graphs (e.g., the class of complete graphs has twin-width zero) and on some classes of sparse graphs (e.g., the class of path graphs has twin-width one), and indeed both graph classes of bounded tree-width and graph classes of bounded rank-width have bounded twin-width [12,24].…”

Section: Introductionmentioning

confidence: 99%

Bounds on the Twin-Width of Product Graphs

Pettersson¹,

Sylvester²

2022

Preprint

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Twin-width is a graph width parameter recently introduced by Bonnet, Kim, Thomassé & Watrigant. Given two graphs G and H and a graph product ⋆, we address the question: is the twin-width of G ⋆ H bounded by a function of the twin-widths of G and H and their maximum degrees? It is known that a bound of this type holds for strong products (Bonnet, Geniet, Kim, Thomassé & Watrigant; SODA 2021).We show that bounds of the same form hold for Cartesian, tensor/direct, rooted, replacement, and zig-zag products. For the lexicographical product we prove that the twin-width of the product of two graphs is exactly the maximum of the twin-widths of the individual graphs. In contrast, for the modular product we show that no bound can hold. In addition, we provide examples showing many of our bounds are tight, and give improved bounds for certain classes of graphs.

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Randomized communication and implicit graph representations

Harms

Wild

Zamaraev

2022

Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

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The most basic lower-bound question in randomized communication complexity is: Does a given problem have constant cost, or non-constant cost? We observe that this question has a deep connection to implicit graph representations in structural graph theory. Specifically, constant-cost communication problems correspond to hereditary graph families that admit constant-size adjacency sketches, or equivalently constant-size probabilistic universal graphs (PUGs), and these graph families are a subset of families that admit adjacency labeling schemes of size 𝑂 (log 𝑛), which are the subject of the well-studied implicit graph question (IGQ).We initiate the study of the hereditary graph families that admit constant-size PUGs, with the two (equivalent) goals of (1) giving a structural characterization of randomized constant-cost communication problems, and (2) resolving a probabilistic version of the IGQ. For each family F studied in this paper (including the monogenic bipartite families, product graphs, interval and permutation graphs, families of bounded twin-width, and others), it holds that the subfamilies H ⊆ F are either stable (in a sense relating to model theory), in which case they admit constant-size PUGs (i.e. adjacency sketches), or they are not stable, in which case they do not.The correspondence between communication problems and hereditary graph families allows for a probabilistic method of constructing adjacency labeling schemes. By this method, we show that the induced subgraphs of any Cartesian products 𝐺 𝑑 are positive examples to the IGQ, also giving a bound on the number of unique induced subgraphs of any graph product. We prove that this probabilistic construction cannot be "naïvely derandomized" by using an Eqality oracle, implying that the Eqality oracle cannot simulate the 𝑘-Hamming Distance communication protocol.As a consequence of our results, we obtain constant-size sketches for deciding dist(𝑥, 𝑦) ≤ 𝑘 for vertices 𝑥, 𝑦 in any stable graph family with bounded twin-width, answering an open question about planar graphs from earlier work. This generalizes to constant-size sketches for deciding first-order formulas over the same graphs.

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Stable graphs of bounded twin-width

Cited by 6 publications

References 14 publications

Sketching Distances in Monotone Graph Classes

Sketching Distances in Monotone Graph Classes

Bounds on the Twin-Width of Product Graphs

Randomized communication and implicit graph representations

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