Dimension and Height for Posets with Planar Cover Graphs

Streib, Noah; Trotter, William T.

doi:10.1016/j.endm.2011.10.035

Cited by 17 publications

(43 citation statements)

References 7 publications

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“…The dimension of orders with a planar diagram respectively a planar cover graph also has received quite some attention. Since this is not central to this note we only provide two recent references where additional pointers can be found: [22] and [32].…”

Section: Order Dimension and Planaritymentioning

confidence: 99%

The Order Dimension of Planar Maps Revisited

Felsner¹

2014

SIAM J. Discrete Math.

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Abstract. Schnyder characterized planar graphs in terms of order dimension. This seminal result found several extensions. A particularly far reaching extension is the Brightwell-Trotter Theorem about planar maps. It states that the order dimension of the incidence poset P M of vertices, edges and faces of a planar map M has dimension at most 4. The original proof generalizes the machinery of Schnyder-paths and Schnyder-regions. In this note we use a simple result about the order dimension of grid intersection graphs to show a slightly stronger result: dim(split(P M )) 4.This may be the rst result in the area that is obtained without using the tools introduced by Schnyder.Mathematics Subject Classications (2010) 05C10, 06A07, 68R10

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Section: Order Dimension and Planaritymentioning

confidence: 99%

The Order Dimension of Planar Maps Revisited

Felsner¹

2014

SIAM J. Discrete Math.

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“…Once this result was published, several researchers in the field expressed their belief that posets with planar cover graphs but bounded height should have bounded dimension. Streib and Trotter showed in that this is indeed the case. Joret et al.…”

Section: Introductionmentioning

confidence: 78%

Topological Minors of Cover Graphs and Dimension

Micek

Wiechert

2017

Journal of Graph Theory

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Abstract. We show that posets of bounded height whose cover graphs exclude a fixed graph as a topological minor have bounded dimension. This result was already proven by Walczak. However, our argument is entirely combinatorial and does not rely on structural decomposition theorems. Given a poset with large dimension but bounded height, we directly find a large clique subdivision in its cover graph. Therefore, our proof is accessible to readers not familiar with topological graph theory, and it allows us to provide explicit upper bounds on the dimension. With the introduced tools we show a second result that is supporting a conjectured generalization of the previous result. We prove that (k + k)-free posets whose cover graphs exclude a fixed graph as a topological minor contain only standard examples of size bounded in terms of k.

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“…However, Streib and Trotter [10] proved that dimension is bounded in terms of height for posets that have a planar cover graph. This stands in sharp contrast with a number of well-known families of posets that have height 2 but unbounded dimension (e.g.…”

Section: Dushnik and Millermentioning

confidence: 99%