Orthogonal Surfaces and Their CP-Orders

Felsner, Stefan; Kappes, Sarah

doi:10.1007/s11083-007-9075-z

Cited by 8 publications

(12 citation statements)

References 28 publications

(38 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Orthogonal structures also open a line for generalizations to higher dimensions. This line of research is the subject of [18]. So far 4 has shown to be more elusive.…”

Section: Order Dimension and Planaritymentioning

confidence: 99%

The Order Dimension of Planar Maps Revisited

Felsner¹

2014

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

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

show abstract

“…Orthogonal structures also open a line for generalizations to higher dimensions. This line of research is the subject of [18]. So far 4 has shown to be more elusive.…”

Section: Order Dimension and Planaritymentioning

confidence: 99%

The Order Dimension of Planar Maps Revisited

Felsner¹

2014

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Schnyder woods were introduced by Schnyder in [35] and [36]. They have numerous applications in the context of graph drawing, e.g., [2,6,28], dimension theory for orders, graphs and polytopes, e.g., [35,8,18], enumeration and encoding of planar structures, e.g., [32,21]. The connection with 3-orientations was found by de Fraysseix and Ossona de Mendez [11].…”

Section: Theorem 83 (See De Fraysseix and De Mendezmentioning

confidence: 99%

Bijections for Baxter families and related objects

Felsner

Fusy

Noy

et al. 2011

Journal of Combinatorial Theory, Series A

Self Cite

View full text Add to dashboard Cite

The Baxter number can be written as $B_n = \sum_0^n \Theta_{k,n-k-1}$. These numbers have first appeared in the enumeration of so-called Baxter permutations; $B_n$ is the number of Baxter permutations of size $n$, and $\Theta_{k,l}$ is the number of Baxter permutations with $k$ descents and $l$ rises. With a series of bijections we identify several families of combinatorial objects counted by the numbers $\Theta_{k,l}$. Apart from Baxter permutations, these include plane bipolar orientations with $k+2$ vertices and $l+2$ faces, 2-orientations of planar quadrangulations with $k+2$ white and $l+2$ black vertices, certain pairs of binary trees with $k+1$ left and $l+1$ right leaves, and a family of triples of non-intersecting lattice paths. This last family allows us to determine the value of $\Theta_{k,l}$ as an application of the lemma of Gessel and Viennot. The approach also allows us to count certain other subfamilies, e.g., alternating Baxter permutations, objects with symmetries and, via a bijection with a class of plan bipolar orientations also Schnyder woods of triangulations, which are known to be in bijection with 3-orientations.Postprint (published version

show abstract

“…, n}. Indeed, Lemma 1 and the fact that it is tight for all point sets that are in general postition and do not lie in a plane containing the all-ones vector can be deduced from a more general theorem of Scarf [14].…”

Section: Proof Of Theoremmentioning

confidence: 96%

Making Octants Colorful and Related Covering Decomposition Problems

Cardinal¹,

Knauer²,

Micek³

et al. 2014

SIAM J. Discrete Math.

View full text Add to dashboard Cite

We give new positive results on the long-standing open problem of geometric covering decomposition for homothetic polygons. In particular, we prove that for any positive integer k, every finite set of points in R 3 can be colored with k colors so that every translate of the negative octant containing at least k 6 points contains at least one of each color. The best previously known bound was doubly exponential in k. This yields, among other corollaries, the first polynomial bound for the decomposability of multiple coverings by homothetic triangles. We also investigate related decomposition problems involving intervals appearing on a line. We prove that no algorithm can dynamically maintain a decomposition of a multiple covering by intervals under insertion of new intervals, even in a semi-online model, in which some coloring decisions can be delayed. This implies that a wide range of sweeping plane algorithms cannot guarantee any bound even for special cases of the octant problem.A preliminary version of this paper was presented at the 2014 ACM-SIAM Symposium on Discrete Algorithms (SODA'14).Journal version of this paper is submitted to SIAM J. Discrete Math.

show abstract

Orthogonal Surfaces and Their CP-Orders

Cited by 8 publications

References 28 publications

The Order Dimension of Planar Maps Revisited

The Order Dimension of Planar Maps Revisited

Bijections for Baxter families and related objects

Making Octants Colorful and Related Covering Decomposition Problems

Contact Info

Product

Resources

About