Coloring and Covering Nowhere Dense Graphs

Grohe, Martin; Kreutzer, Stephan; Rabinovich, Roman; Siebertz, Sebastian; Stavropoulos, Konstantinos S.

doi:10.1137/18m1168753

Cited by 33 publications

(27 citation statements)

References 13 publications

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“…Another attractive aspect of strong colouring numbers is that they interpolate between degeneracy and treewidth [40]. As previously noted, scol 1 (G) equals the degeneracy of G plus 1.…”

Section: Generalised Colouring Numbersmentioning

confidence: 99%

“…As previously noted, scol 1 (G) equals the degeneracy of G plus 1. At the other extreme, Grohe et al [40] showed that scol s (G) tw(G) + 1 for every integer s 1, and indeed lim…”

Section: Generalised Colouring Numbersmentioning

confidence: 99%

“…Colouring numbers are important because they characterise bounded expansion [78] and nowhere dense classes [40], and have several algorithmic applications [32,41].…”

Section: Generalised Colouring Numbersmentioning

confidence: 99%

“…For weak colouring numbers, Grohe et al [40] showed that if H is an r-topological minor of G then wcol s (H) 2 wcol 2rs+s (G) for every integer s 1. The following lemma extends this result to shallow minors.…”

Section: Generalised Colouring Numbersmentioning

confidence: 99%

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Shallow Minors, Graph Products and Beyond Planar Graphs

Hickingbotham¹,

Wood²

2021

Preprint

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The planar graph product structure theorem of Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] states that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. This result has been the key tool to resolve important open problems regarding queue layouts, nonrepetitive colourings, centered colourings, and adjacency labelling schemes. In this paper, we extend this line of research by utilizing shallow minors to prove analogous product structure theorems for several beyond planar graph classes. The key observation that drives our work is that many beyond planar graphs can be described as a shallow minor of the strong product of a planar graph with a small complete graph. In particular, we show that power of planar graphs, k-planar, (k, p)-cluster planar, k-semi-fan-planar graphs and k-fan-bundle planar graphs can be described in this manner. Using a combination of old and new results, we deduce that these classes have bounded queuenumber, bounded nonrepetitive chromatic number, polynomial p-centred chromatic numbers, linear strong colouring numbers, and cubic weak colouring numbers. In addition, we show that k-gap planar graphs have super-linear local treewidth and, as a consequence, cannot be described as a subgraph of the strong product of a graph with bounded treewidth and a path.

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“…Another attractive aspect of strong colouring numbers is that they interpolate between degeneracy and treewidth [40]. As previously noted, scol 1 (G) equals the degeneracy of G plus 1.…”

Section: Generalised Colouring Numbersmentioning

confidence: 99%

“…As previously noted, scol 1 (G) equals the degeneracy of G plus 1. At the other extreme, Grohe et al [40] showed that scol s (G) tw(G) + 1 for every integer s 1, and indeed lim…”

Section: Generalised Colouring Numbersmentioning

confidence: 99%

“…Colouring numbers are important because they characterise bounded expansion [78] and nowhere dense classes [40], and have several algorithmic applications [32,41].…”

Section: Generalised Colouring Numbersmentioning

confidence: 99%

Section: Generalised Colouring Numbersmentioning

confidence: 99%

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Shallow Minors, Graph Products and Beyond Planar Graphs

Hickingbotham¹,

Wood²

2021

Preprint

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“…Let G ′ be the 6 tw(G)-subdivision of a graph G. Let G := {G ′ : G is a graph}. Grohe, Kreutzer, Rabinovich, Siebertz, and Stavropoulos [29] proved that G has linear expansion. On the other hand, Dubois, Joret, Perarnau, Pilipczuk, and Pitois [11] proved that there are graphs G such that every p-centred colouring of G ′ has at least 2 cp 1/2 colours, for some constant c > 0.…”

Section: Open Problem 31 Is There a Proof Of Theorem 3 Or Theorem 4 mentioning

confidence: 99%

Notes on Graph Product Structure Theory

et al. 2021

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It was recently proved that every planar graph is a subgraph of the strong product of a path and a graph with bounded treewidth. This paper surveys generalisations of this result for graphs on surfaces, minor-closed classes, various non-minor-closed classes, and graph classes with polynomial growth. We then explore how graph product structure might be applicable to more broadly defined graph classes. In particular, we characterise when a graph class defined by a cartesian or strong product has bounded or polynomial expansion. We then explore graph product structure theorems for various geometrically defined graph classes, and present several open problems.

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On weighted sublinear separators

Dvořák

2021

Journal of Graph Theory

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Coloring and Covering Nowhere Dense Graphs

Cited by 33 publications

References 13 publications

Shallow Minors, Graph Products and Beyond Planar Graphs

Shallow Minors, Graph Products and Beyond Planar Graphs

Notes on Graph Product Structure Theory

On weighted sublinear separators

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