Структурная Разреженность

Нешетрил, Ярослав; Nešetřil, Jaroslav; Мендез, Патрис Оссона де; Mendez, Patrice Ossona de

doi:10.4213/rm9688

Cited by 4 publications

(1 citation statement)

References 69 publications

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“…Natural graph parameters that have played a central role in the structural theory of tournaments developed by Chudnovsky, Seymour and others [12,13,14,15,18] are cut width and path width. The more recent theory of structural sparsity [19,20,35,36] highlights clique width and twin width. Here twin width is the key parameter.…”

Section: Introductionmentioning

confidence: 99%

Isomorphism Testing for Graphs Excluding Small Minors

2023

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We prove that isomorphism of tournaments of twin width at most k can be decided in time k O(log k) n O(1) . This implies that the isomorphism problem for classes of tournaments of bounded or moderately growing twin width is in polynomial time. By comparison, there are classes of undirected graphs of bounded twin width that are isomorphism complete, that is, the isomorphism problem for the classes is as hard as the general graph isomorphism problem. Twin width is a graph parameter that has been introduced only recently (Bonnet et al., FOCS 2020), but has received a lot of attention in structural graph theory since then. On directed graphs, it is functionally smaller than clique width. We prove that on tournaments (but not on general directed graphs) it is also functionally smaller than directed tree width (and thus, the same also holds for cut width and directed path width). Hence, our result implies that tournament isomorphism testing is also fixed-parameter tractable when parameterized by any of these parameters.Our isomorphism algorithm heavily employs group-theoretic techniques. This seems to be necessary: as a second main result, we show that the combinatorial Weisfeiler-Leman algorithm does not decide isomorphism of tournaments of twin width at most 35 if its dimension is o( √ n). (Throughout this abstract, n is the order of the input graphs.

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Section: Introductionmentioning

confidence: 99%

Isomorphism Testing for Graphs Excluding Small Minors

2023

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Nowhere Dense Graph Classes and Dimension

et al. 2019

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Nowhere dense graph classes provide one of the least restrictive notions of sparsity for graphs. Several equivalent characterizations of nowhere dense classes have been obtained over the years, using a wide range of combinatorial objects. In this paper we establish a new characterization of nowhere dense classes, in terms of poset dimension: A monotone graph class is nowhere dense if and only if for every h 1 and every ε > 0, posets of height at most h with n elements and whose cover graphs are in the class have dimension O(n ε ).

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