Nowhere Dense Graph Classes and Dimension

Joret, Gwenaël; Micek, Piotr; Mendez, Patrice Ossona de; Wiechert, Veit

doi:10.1007/s00493-019-3892-8

Cited by 11 publications

(12 citation statements)

References 38 publications

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“…Joret, Micek and Wiechert [17] have recently shown that for fixed t ≥ 3, d(t, h) grows exponentially with h. The best bound to date in the general case is due to Joret, Micek, Ossona de Mendez and Wiechert [16], where they prove:…”

Section: Connections With Structural Graph Theorymentioning

confidence: 98%

“…However, an exponential upper bound was given in [16], and more recently, two groups have announced a polynomial upper bound on c h . Joret, Micek, Ossona de Mendez and Wiechert have shown how their results in [16] can be extended to obtain this conclusion. Meanwhile, Kozik, Krawczyk, Micek and Trotter [24] have a much more complicated argument which [17] showed that the c h ≥ 2h − 2.…”

Section: Planar Posets and Dimensionmentioning

confidence: 99%

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Comparing Dushnik-Miller Dimension, Boolean Dimension and Local Dimension

Barrera-Cruz

Prag

Smith

et al. 2019

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The original notion of dimension for posets is due to Dushnik and Miller and has been studied extensively in the literature. Quite recently, there has been considerable interest in two variations of dimension known as Boolean dimension and local dimension. For a poset P , the Boolean dimension of P and the local dimension of P are both bounded from above by the dimension of P and can be considerably less. Our primary goal will be to study analogies and contrasts among these three parameters. As one example, it is known that the dimension of a poset is bounded as a function of its height and the tree-width of its cover graph. The Boolean dimension of a poset is bounded in terms of the tree-width of its cover graph, independent of its height. We show that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of height. We also prove that the local dimension of a poset is bounded in terms of the path-width of its cover graph. In several of our results, Ramsey theoretic methods will be applied.

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Section: Connections With Structural Graph Theorymentioning

confidence: 98%

Section: Planar Posets and Dimensionmentioning

confidence: 99%

Comparing Dushnik-Miller Dimension, Boolean Dimension and Local Dimension

Barrera-Cruz

Prag

Smith

et al. 2019

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“…Thus dim(P ) is the least positive integer d such that P has a realizer of size d. Accordingly, to establish an upper bound of the form dim(P ) d, the most natural approach is simply to construct a realizer of size d for P . However, in recent papers [8,9,12,13,14,15,16,21,23,28,30], another approach has been taken. Let Inc(P ) denote the set of ordered incomparable pairs of P .…”

Section: Notation Terminology and Background Materialsmentioning

confidence: 99%

“…Recently, a number of important results connecting dimension with structural graph theory have been proved [2,12,13,14,16,21,30]. In particular, the following generalization of Theorem 1.3 is proved in [30] (see [21] for an alternative proof and [13,16] for further extensions). Theorem 7.1.…”

Section: Connections With Graph Minorsmentioning

confidence: 99%

Dimension of posets with planar cover graphs excluding two long incomparable chains

Howard¹,

Streib²,

Trotter

et al. 2019

Journal of Combinatorial Theory, Series A

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It has been known for more than 40 years that there are posets with planar cover graphs and arbitrarily large dimension. Recently, Streib and Trotter proved that such posets must have large height. In fact, all known constructions of such posets have two large disjoint chains with all points in one chain incomparable with all points in the other. Gutowski and Krawczyk conjectured that this feature is necessary. More formally, they conjectured that for every k 1, there is a constant d such that if P is a poset with a planar cover graph and P excludes k + k, then dim(P ) d. We settle their conjecture in the affirmative. We also discuss possibilities of generalizing the result by relaxing the condition that the cover graph is planar. arXiv:1608.08843v2 [math.CO]

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“…It was proved by Joret, Micek, Ossona de Mendez and Wiechert [12] that if P is a poset of height at most h, and the cover graph of P has weak (3h − 3)-coloring number at most k, then dim(P ) 4 k . For posets P of height h = 2, this implies that the dimension is at most 4 k , where k is the weak 3-coloring number of the comparability graph of P. This will be significantly improved in Section 2 (see Theorem 4).…”

Section: Introductionmentioning

confidence: 99%

Boxicity, Poset Dimension, and Excluded Minors

Wiechert

2018

Electron. J. Combin.

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In this short note, we relate the boxicity of graphs (and the dimension of posets) with their generalized coloring parameters. In particular, together with known estimates, our results imply that any graph with no $K_t$-minor can be represented as the intersection of $O(t^2\log t)$ interval graphs (improving the previous bound of $O(t^4)$), and as the intersection of $\tfrac{15}2 t^2$ circular-arc graphs.

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Nowhere Dense Graph Classes and Dimension

Cited by 11 publications

References 38 publications

Comparing Dushnik-Miller Dimension, Boolean Dimension and Local Dimension

Comparing Dushnik-Miller Dimension, Boolean Dimension and Local Dimension

Dimension of posets with planar cover graphs excluding two long incomparable chains

Boxicity, Poset Dimension, and Excluded Minors

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