Comparing Dushnik-Miller Dimension, Boolean Dimension and Local Dimension

Barrera-Cruz, Fidel; Prag, Thomas; Smith, Heather; Taylor, Libby; Trotter, William T.

doi:10.1007/s11083-019-09502-6

Cited by 13 publications

(20 citation statements)

References 38 publications

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“…The following lemma is analogous to a lemma proved for local dimension by Barrera-Cruz, Prag, Smith, Taylor, and Trotter in [1].…”

Section: Local 2-dimension and Complete Bipartite Edge-coverings Of G...mentioning

confidence: 62%

Local $t$-dimension

Lewis

2020

Preprint

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“…The following lemma is analogous to a lemma proved for local dimension by Barrera-Cruz, Prag, Smith, Taylor, and Trotter in [1].…”

Section: Local 2-dimension and Complete Bipartite Edge-coverings Of G...mentioning

confidence: 62%

Local $t$-dimension

Lewis

2020

Preprint

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“…In [1], it is shown that for each d 1, there is a poset P with bdim(P ) 4 and ldim(P ) > d. Our construction provides another instance of a family of posets where Boolean dimension is bounded and local dimension is not.…”

Section: Bounded Boolean Dimension and Unbounded Local Dimensionmentioning

confidence: 96%

“…Barrera-Cruz, Prag, Smith, Taylor and Trotter [1] proved that the local dimension of a poset is bounded in terms of the path-width of its cover graph, independent of its height. Formally, here is their result.…”

Section: Structural Graph Theorymentioning

confidence: 99%

“…However, it is also shown in [1] that the analogue of Theorem 5 for local dimension is false: Theorem 7. For every d…”

Section: Structural Graph Theorymentioning

confidence: 99%

“…Actually, we will only use the case where k = 1, and now the theorem becomes a multidimensional version of the pigeon-hole principle. Readers who would be interested in how this theorem is applied to combinatorial problems on posets when k 2 are encouraged to consult [1], [4] and [17]. Now we are ready to state and prove our main theorem.…”

Section: The Local Dimension Of Posets In the Class Kmentioning

confidence: 99%

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Local Dimension is Unbounded for Planar Posets

Bosek¹,

Grytczuk²,

Trotter³

2020

Electron. J. Combin.

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In 1981, Kelly showed that planar posets can have arbitrarily large dimension. However, the posets in Kelly's example have bounded Boolean dimension and bounded local dimension, leading naturally to the questions as to whether either Boolean dimension or local dimension is bounded for the class of planar posets. The question for Boolean dimension was first posed by Nešetřil and Pudlák in 1989 and remains unanswered today. The concept of local dimension is quite new, introduced in 2016 by Ueckerdt. Since that time, researchers have obtained many interesting results concerning Boolean dimension and local dimension, contrasting these parameters with the classic Dushnik-Miller concept of dimension, and establishing links between both parameters and structural graph theory, path-width and tree-width in particular. Here we show that local dimension is not bounded on the class of planar posets. Our proof also shows that the local dimension of a poset is not bounded in terms of the maximum local dimension of its blocks, and it provides an alternative proof of the fact that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of its height.

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Local and Union Page Numbers

Merker

Ueckerdt

2019

Lecture Notes in Computer Science

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We introduce the novel concepts of local and union book embeddings, and, as the corresponding graph parameters, the local page number pn (G) and the union page number pn u (G). Both parameters are relaxations of the classical page number pn(G), and for every graph G we have pn (G) pn u (G) pn(G). While for pn(G) one minimizes the total number of pages in a book embedding of G, for pn (G) we instead minimize the number of pages incident to any one vertex, and for pn u (G) we instead minimize the size of a partition of G with each part being a vertex-disjoint union of crossing-free subgraphs. While pn (G) and pn u (G) are always within a multiplicative factor of 4, there is no bound on the classical page number pn(G) in terms of pn (G) or pn u (G). We show that local and union page numbers are closer related to the graph's density, while for the classical page number the graph's global structure can play a much more decisive role. We introduce tools to investigate local and union book embeddings in exemplary considerations of the class of all planar graphs and the class of graphs of tree-width k.As an incentive to pursue research in this new direction, we offer a list of intriguing open problems.

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Comparing Dushnik-Miller Dimension, Boolean Dimension and Local Dimension

Cited by 13 publications

References 38 publications

Local $t$-dimension

Local $t$-dimension

Local Dimension is Unbounded for Planar Posets

Local and Union Page Numbers

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