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Counting and Enumerating Pointed Pseudotriangulations with the Greedy Flip Algorithm

Hervé Brönnimann, Lutz Kettner, Michel Pocchiola, Jack Snoeyink
2006 SIAM journal on computing (Print)  
We present an algorithm to enumerate the pseudo-triangulations of a given point set, based on the greedy flip algorithm of Pocchiola and Vegter [Topologically sweeping visibility complexes via pseudo-triangulations  ...  Bespamyatnikh has extended his enumeration algorithm [14] to pseudo-triangulations, but has yet to implement it.  ...  Acknowledgement We thank Ileana Streinu and the other participants of the NSF-supported Bellairs workshop on pseudo-triangulations for many enjoyable discussions.  ... 
doi:10.1137/050631008 fatcat:6t4t5isr5zg7zacem7pr5rcpoe

Flips in planar graphs

Prosenjit Bose, Ferran Hurtado
2009 Computational geometry  
We review results concerning edge flips in planar graphs concentrating mainly on various aspects of the following problem: Given two different planar graphs of the same size, how many edge flips are necessary  ...  and sufficient to transform one graph into another?  ...  Notice that the enumeration of all the distinct planar graphs of a given size and type amounts to the enumeration of the vertices of the flip graph.  ... 
doi:10.1016/j.comgeo.2008.04.001 fatcat:fxq5ql3imzftpoh5naidvy5e3q

Product-Coproduct Prographs and Triangulations of the Sphere [article]

Nicolas Borie, Justine Falque
2022 arXiv   pre-print
In this paper, we explain how the classical Catalan families of objects involving paths, tableaux, triangulations, parentheses configurations and more generalize canonically to a three-dimensional version  ...  In particular, we present product-coproduct prographs as central objects explaining the combinatorics of the triangulations of the sphere.  ...  Indeed, prographs are planar assemblies of operators with identified inputs and outputs; if one puts reasonable operators on the vertices of a planar oriented map, with respect to the number of ongoing  ... 
arXiv:2202.05757v1 fatcat:ixcideilgzdgla2hb7hwlntheq

Page 5319 of Mathematical Reviews Vol. , Issue 2000h [page]

2000 Mathematical Reviews  
From the summary: “We enumerate two families of rooted planar near-triangulations with respect to the number of flippable edges. It is shown that their generating functions are algebraic.  ...  diagonal flip; e is flippable if flipping it does not create a loop, i.e. if c £d; in a triangulation, one requires additionally that the flipping does not create multiple edges.  ... 

Genus dependence of the number of (non-)orientable surface triangulations

Benedikt Krüger, Klaus Mecke
2016 Physical Review D  
We verify that the limit of the entropy density of triangulations is independent of genus and orientability and are able to determine the next-to-leading and the next-to-next-to-leading order terms.  ...  Topological triangulations of orientable and non-orientable surfaces with arbitrary genus have important applications in quantum geometry, graph theory and statistical physics.  ...  Benedetti for calling our attention to Ref. [50] . This work is supported by EFI Quantum Geometry and the Elite Network of Bavaria.  ... 
doi:10.1103/physrevd.93.085018 fatcat:7j56lw4lxvbkjnzfdvhcqpl6oq

Cluster algebras of type D: pseudotriangulations approach [article]

Cesar Ceballos, Vincent Pilaud
2015 arXiv   pre-print
Finally, we discuss applications of our model to polytopal realizations of type D associahedra and connections to subword complexes and c-cluster complexes.  ...  We present a combinatorial model for cluster algebras of type D_n in terms of centrally symmetric pseudotriangulations of a regular 2n-gon with a small disk in the centre.  ...  For example, for a cluster variable y and an initial cluster seed X in type A, the denominator of y with respect to X is the product of the variables in X whose diagonals cross the diagonal y, while the  ... 
arXiv:1504.06377v1 fatcat:d2jw2n6xxrcuxpy4zw7wp5yoiu

Cluster Algebras of Type D: Pseudotriangulations Approach

Cesar Ceballos, Vincent Pilaud
2015 Electronic Journal of Combinatorics  
Finally, we discuss applications of our model to polytopal realizations of type $D$ associahedra and connections to subword complexes and $c$-cluster complexes.  ...  We present a combinatorial model for cluster algebras of type $D_n$ in terms of centrally symmetric pseudotriangulations of a regular $2n$ gon with a small disk in the centre.  ...  For example, for a cluster variable y and an initial cluster seed X in type A, the denominator of y with respect to X is the product of the variables in X whose diagonals cross the diagonal y, while the  ... 
doi:10.37236/5282 fatcat:uv5c42zh5ncehkkjlnzab7ku3u

Page 6813 of Mathematical Reviews Vol. , Issue 2000j [page]

2000 Mathematical Reviews  
Our proof is based on the fact that a 3-connected graph admits an ear assembly having some special properties with respect to the non- separating cycles of the graph.  ...  {For the entire collection see MR 2000h:00011.} 2000j:05039 05C10 Watanabe, Takahiro (J- YOKOTEH; Yokohama); Negami, Seiya (J-YOKOTEH; Yokohama) Diagonal flips in pseudo-triangulations on closed surfaces  ... 

On the mixing time of the flip walk on triangulations of the sphere [article]

Thomas Budzinski
2018 arXiv   pre-print
A simple way to sample a uniform triangulation of the sphere with a fixed number n of vertices is a Monte-Carlo method: we start from an arbitrary triangulation and flip repeatedly a uniformly chosen edge  ...  We give a lower bound in n^5/4 on the mixing time of this Markov chain.  ...  We say that flip(t, e) is obtained from t by flipping the edge e (cf. Figure 1 ). Note that it is possible to flip a loop and to flip the root edge.  ... 
arXiv:1611.07324v2 fatcat:zda22xyhr5h6vkykm742vaf33q

Enumerating Constrained Non-crossing Minimally Rigid Frameworks

David Avis, Naoki Katoh, Makoto Ohsaki, Ileana Streinu, Shin-ichi Tanigawa
2007 Discrete & Computational Geometry  
In particular, we obtain that the set of all the constrained non-crossing Laman frameworks on a given point set is connected by flips which restore the Laman property.  ...  Laman frameworks) on a given generic set of n points.  ...  Relevant to the historical context of our work are the results of Bereg [7, 10] using reverse search combined with data-specific lexicographic orderings to enumerate triangulations and pointed pseudo-triangulations  ... 
doi:10.1007/s00454-007-9026-x fatcat:lrjbzrrcrrdahkopq7d7jf26du

Short Encodings of Evolving Structures

Daniel D. Sleator, Robert E. Trajan, William P. Thurston
1992 SIAM Journal on Discrete Mathematics  
Similarly, it is shown that n(n1ogn) "diagonal flips" are required in the worst case to transform one n-vertex numbered triangulated planar graph into some other one.  ...  An O(n log n) upper bound for associative, commutative operations was known previously, whereas here an O(n log n) upper bound for diagonal flips is obtained. 1. Introduction.  ...  respect to 1).  ... 
doi:10.1137/0405034 fatcat:geacnzwjh5amlnfkhggswigf3a

Pseudo-Triangulations - a Survey [article]

Guenter Rote, Francisco Santos, Ileana Streinu
2007 arXiv   pre-print
A pseudo-triangle is a simple polygon with three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles.  ...  Pseudo-triangulations appear as data structures in computational geometry, as planar bar-and-joint frameworks in rigidity theory and as projections of locally convex surfaces.  ...  The diagonal e divides R into two polygons R ′ and R ′′ with k ′ and k ′′ corners respectively, with k ′ +k ′′ = k+2.  ... 
arXiv:math/0612672v2 fatcat:adhyppd3wjbhnfmdky5dbxjxzu

Signed permutations and the four color theorem [article]

Shalom Eliahou, Cedric Lecouvey
2006 arXiv   pre-print
To each permutation σ in S_n we associate a triangulation of a fixed (n+2)-gon.  ...  We then determine the fibers of this association and show that they coincide with the sylvester classes depicted By Novelli, Hivert and Thibon.  ...  We suppose in the sequel that we have chosen a labelling of the n(n+1) 2 diagonals of P (for example we can consider the labelling by roots belonging to the root system of type A n−1 depicted in [3] )  ... 
arXiv:math/0606726v1 fatcat:ul4hyxijunhb7hi3znjzkeawi4

On the expected number of perfect matchings in cubic planar graphs [article]

Marc Noy, Clément Requilé, Juanjo Rué
2021 arXiv   pre-print
Our starting point is a correspondence between counting perfect matchings in rooted cubic planar maps and the partition function of the Ising model in rooted triangulations.  ...  In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with n vertices.  ...  The third author acknowledges preliminary discussions on this topic with Mihyun Kang, Michael Mosshammer and Philipp Sprüssel during a visit to the Technical University of Graz in 2015, and wishes to thank  ... 
arXiv:2005.13821v2 fatcat:vrqwkhpdnver7hhzzizl63pho4

Counting plane graphs: Perfect matchings, spanning cycles, and Kasteleyn's technique

Micha Sharir, Adam Sheffer, Emo Welzl
2013 Journal of combinatorial theory. Series A  
More specifically, we bound the ratio between the number of spanning cycles (or perfect matchings) that can be embedded over a point set and the number of triangulations that can be embedded over it.  ...  The respective bounds are O (1.8181 N ) for cycles and O (1.1067 N ) for matchings.  ...  Acknowledgments We would like to thank the two anonymous referees for their helpful comments.  ... 
doi:10.1016/j.jcta.2013.01.002 fatcat:u62vgjo7o5azthn7qbwjybgixm
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