Straight-Line Drawing of Quadrangulations

Fusy, Éric

doi:10.1007/978-3-540-70904-6_23

Cited by 9 publications

(6 citation statements)

References 9 publications

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“…In comparison, the best previously known algorithm [22] only guarantees a grid size (⌈n/2⌉ − 1) × ⌊n/2⌋. The principle of our straight-line drawing algorithm has also been recently applied to the family of quadrangulations [15,16]; for a random quadrangulation with n vertices, the grid size is with high probability 13n/27 × 13n/27 up to an additive error O( √ n), which improves on the grid size (⌈n/2⌉ − 1) × ⌊n/2⌋ guaranteed by [2].…”

Section: Discussionmentioning

confidence: 99%

Transversal structures on triangulations: A combinatorial study and straight-line drawings

Fusy

2009

Discrete Mathematics

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111

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This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, which are called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edge-labelling and consists of two bipolar orientations that are transversal. For this reason, the terminology used here is that of transversal structures. The main results obtained in the article are a bijection between irreducible triangulations and ternary trees, and a straight-line drawing algorithm for irreducible triangulations. For a random irreducible triangulation with n vertices, the grid size of the drawing is asymptotically with high probability 11n/27 × 11n/27 up to an additive error of O( √ n). In contrast, the best previously known algorithm for these triangulations only guarantees a grid size (⌈n/2⌉ − 1) × ⌊n/2⌋.

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Section: Discussionmentioning

confidence: 99%

Transversal structures on triangulations: A combinatorial study and straight-line drawings

Fusy

2009

Discrete Mathematics

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111

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“…Plane bipolar orientations yield geometric representations of graphs in various flavors (visibility [39], floor planning [33,27], straight-line drawing [40,20]). The thesis of Ossona de Mendez is devoted to studying their beautiful properties and applications [14]; see also [13] for a detailed survey.…”

Section: Maps Quadrangulations and Orientationsmentioning

confidence: 99%

“…Bipolar orientations have proved to be insightful in solving many algorithmic problems such as planar graph embedding [31,11] and geometric representations of graphs in various flavors (visibility [41], floor planning [35,30], straight-line drawing [42,23]). They also constitute a beautiful combinatorial structure; the thesis of Ossona de Mendez is devoted to studying their numerous properties and applications [17]; see also [16] for a detailed survey.…”

Section: Notementioning

confidence: 99%

Bijections for Baxter families and related objects

Felsner

Fusy

Noy

et al. 2011

Journal of Combinatorial Theory, Series A

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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

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“…This is a polynomial expression of degree four, homogeneous in the variables i+1, j +1. Additionally, discrete harmonic functions seem promising for the random generation of random walks confined to cones [21].…”

Section: Discrete Harmonic Functionsmentioning

confidence: 99%

Weighted lattice walks and universality classes

Courtiel¹,

Melczer²,

Mishna³

et al. 2017

Journal of Combinatorial Theory, Series A

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In this work we consider two different aspects of weighted walks in cones. To begin we examine a particular weighted model, known as the Gouyou-Beauchamps model. Using the theory of analytic combinatorics in several variables we obtain the asymptotic expansion of the total number of Gouyou-Beauchamps walks confined to the quarter plane. Our formulas are parametrized by weights and starting point, and we identify six different asymptotic regimes (called universality classes) which arise according to the values of the weights. The weights allowed in this model satisfy natural algebraic identities permitting an expression of the weighted generating function in terms of the generating function of unweighted walks on the same steps. The second part of this article explains these identities combinatorially for walks in arbitrary cones and dimensions, and provides a characterization of universality classes for general weighted walks. Furthermore, we describe an infinite set of models with non-D-finite generating function.

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Straight-Line Drawing of Quadrangulations

Cited by 9 publications

References 9 publications

Transversal structures on triangulations: A combinatorial study and straight-line drawings

Transversal structures on triangulations: A combinatorial study and straight-line drawings

Bijections for Baxter families and related objects

Weighted lattice walks and universality classes

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