Dimension and height for posets with planar cover graphs

Streib, Noah; Trotter, William T.

doi:10.1016/j.ejc.2013.06.017

Cited by 28 publications

(36 citation statements)

References 14 publications

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“…In 2014, Streib and Trotter [30] proved that for every positive integer h, there is a least positive integer c h so that if P is a poset of height h and the cover graph of P is planar, then dim(P ) ≤ c h . The proof given in [30] merely established the existence of c h and gave no useful information about its size. However, an exponential upper bound was given in [16], and more recently, two groups have announced a polynomial upper bound on c h .…”

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: Planar Posets and Dimensionmentioning

confidence: 99%

“…Using duality, analogous statements hold for maximal elements. Note, however, that there are no statements of this type for posets with planar cover graphs, since as pointed out in [30], for every d ≥ 1, there is a poset P with a zero and a one such that dim(P ) ≥ d and the cover graph of P is planar.…”

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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“…On the other hand, Trotter and Moore [9] proved that they are the only obstructions for a set of incomparable pairs to be reversible, i.e., S ⊆ Inc(P ) is reversible if and only if it does not contain an alternating cycle. For more details about now standard concepts and techniques for working with Dushnik-Miller dimension, the reader may consult any of several recent research papers, e.g., [4], [7] and [11] or the research monograph [8].…”

Section: Notation and Terminologymentioning

confidence: 99%

Boolean Dimension, Components and Blocks

Mészáros

Micek

Trotter

2019

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We investigate the behavior of Boolean dimension with respect to components and blocks. To put our results in context, we note that for Dushnik-Miller dimension, we have that if dim(C) ≤ d for every component C of a poset P , then dim(P ) ≤ max{2, d}; also if dim(B) ≤ d for every block B of a poset P , then dim(P ) ≤ d + 2. By way of constrast, local dimension is well behaved with respect to components, but not for blocks: if ldim(C) ≤ d for every component C of a poset P , then ldim(P ) ≤ d + 2; however, for every d ≥ 4, there exists a poset P with ldim(P ) = d and dim(B) ≤ 3 for every block B of P . In this paper we show that Boolean dimension behaves like Dushnik-Miller dimension with respect to both components and blocks: if bdim(C) ≤ d for every component C of P , then bdim(P ) ≤ 2 + d + 4 · 2 d ; also if bdim(B) ≤ d for every block of P , then bdim(P ) ≤ 19 + d + 18 · 2 d .

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“…This result is the latest step in a series of recent works connecting poset dimension with graph structure theory. This line of research began with the following result of Streib and Trotter [26]: For every fixed h 1, posets of height h with a planar cover graph have bounded dimension. That is, the dimension of posets with planar cover graphs is bounded from above by a function of their height.…”

Section: Introductionmentioning

confidence: 99%

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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Dimension and height for posets with planar cover graphs

Cited by 28 publications

References 14 publications

Comparing Dushnik-Miller Dimension, Boolean Dimension and Local Dimension

Comparing Dushnik-Miller Dimension, Boolean Dimension and Local Dimension

Boolean Dimension, Components and Blocks

Nowhere Dense Graph Classes and Dimension

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