Asymptotic dimension of planes and planar graphs

Fujiwara, Koji; Papasoglu, Panos

doi:10.48550/arxiv.2002.01630

Cited by 5 publications

(10 citation statements)

References 20 publications

(16 reference statements)

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“…Note that trees have tree-width at most 1 and cacti have tree-width at most 2. So Theorems 1.3 and 1.4 are optimal and generalize the results of Bell and Dranishnikov [4] and Fujiwara and Papasoglu [10]. Theorems 1.3 and 1.4 also generalize results of Bonamy et al [5] who proved the same for the class of graphs of bounded path-width and proved the case with the extra bounded maximum degree condition.…”

Section: Introductionsupporting

confidence: 72%

“…The work of this paper is motivated by the following question of Fujiwara and Papasoglu [10] about asymptotic dimension and graph minors. A graph H is a minor of another graph G if H is isomorphic to a graph that can be obtained from a subgraph of G by contracting edges.…”

Section: Introductionmentioning

confidence: 99%

“…For every positive integer w, the asymptotic dimension of the class of graphs of tree-width at most w is at most 1. [4] and Fujiwara and Papasoglu [10], respectively, showed that the class of trees and the class of cacti, respectively, have asymptotic dimension 1. Note that trees have tree-width at most 1 and cacti have tree-width at most 2.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic dimension of minor-closed families and beyond

Liu

2020

Preprint

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The asymptotic dimension of metric spaces is an important notion in geometric group theory introduced by Gromov. The metric spaces considered in this paper are the ones whose underlying spaces are the vertex-sets of graphs and whose metrics are the distance functions in graphs. A standard compactness argument shows that it suffices to consider the asymptotic dimension of classes of finite graphs.In this paper we prove that the asymptotic dimension of any minor-closed family, any class of graphs of bounded tree-width, and any class of graphs of bounded layered tree-width are at most 2, 1, and 2, respectively. The first result solves a question of Fujiwara and Papasoglu; the second and third results solve a number of questions of Bonamy, Bousquet, Esperet, Groenland, Pirot and Scott. These bounds for asymptotic dimension are optimal and improve a number of results in the literature. Our proofs can be transformed into linear or quadratic time algorithms for finding coverings witnessing the asymptotic dimension which is equivalent to finding weak diameter colorings for graphs. The key ingredient of our proof is a unified machinery about the asymptotic dimension of classes of graphs that have tree-decompositions of bounded adhesion over hereditary classes with known asymptotic dimension, which might be of independent interest.

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Section: Introductionsupporting

confidence: 72%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Asymptotic dimension of minor-closed families and beyond

Liu

2020

Preprint

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“…Asymptotic dimension is relevant in several contexts: in geometric group theory, as groups of finite asymptotic dimension satisfy the Novikov conjecture [21], in geometry [17], [7] and in graph theory [22], [11], [4].…”

Section: Introductionmentioning

confidence: 99%

“…From the geometric point of view it is interesting to calculate the exact asymptotic dimension of a space. This has been accomplished for several 'natural' classes of spaces: It is shown in [7] that the asymptotic dimension of a hyperbolic group G is equal to dim(∂G) + 1, it is shown in [12], [17] that n-dimensional Hadamard manifolds of pinched negative curvature have asymptotic dimension n, and in [11], [15], [4] that planar graphs (or more generally planar geodesic metric spaces) have asymptotic dimension at most 2. In this paper we extend this list to the class of spaces with polynomial growth.…”

Section: Introductionmentioning

confidence: 99%

Polynomial growth and asymptotic dimension

Papasoglu¹

2021

Preprint

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Asymptotic dimension of minor-closed families and beyond

Liu¹

2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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The asymptotic dimension of metric spaces is an important notion in geometric group theory introduced by Gromov. The metric spaces considered in this paper are the ones whose underlying spaces are the vertex-sets of graphs and whose metrics are the distance functions in graphs. A standard compactness argument shows that it suffices to consider the asymptotic dimension of classes of finite graphs.In this paper we prove that the asymptotic dimension of any proper minor-closed family, any class of graphs of bounded tree-width, and any class of graphs of bounded layered tree-width are at most 2, 1, and 2, respectively. The first result solves a question of Fujiwara and Papasoglu; the second and third results solve a number of questions of Bonamy, Bousquet, Esperet, Groenland, Pirot and Scott. These bounds for asymptotic dimension are optimal and improve a number of results in the literature. Our proofs can be transformed into linear or quadratic time algorithms for finding coverings witnessing the asymptotic dimension which is equivalent to finding weak diameter colorings for graphs.The key ingredient of our proof is a unified machinery about the asymptotic dimension of classes of graphs that have tree-decompositions of bounded adhesion over hereditary classes with known asymptotic dimension, which might be of independent interest.

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Asymptotic dimension of planes and planar graphs

Abstract: We show that the asymptotic dimension of a geodesic space that is homeomorphic to a subset in the plane is at most three. In particular, the asymptotic dimension of the plane and any planar graph is at most three.

Cited by 5 publications

References 20 publications

Asymptotic dimension of minor-closed families and beyond

Asymptotic dimension of minor-closed families and beyond

Polynomial growth and asymptotic dimension

Asymptotic dimension of minor-closed families and beyond

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