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Counting and Enumerating Pointed Pseudotriangulations with the Greedy Flip Algorithm

2006
*
SIAM journal on computing (Print)
*

We present an algorithm

doi:10.1137/050631008
fatcat:6t4t5isr5zg7zacem7pr5rcpoe
*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. ...##
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Flips in planar graphs

2009
*
Computational geometry
*

We review results concerning edge

doi:10.1016/j.comgeo.2008.04.001
fatcat:fxq5ql3imzftpoh5naidvy5e3q
*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. ...##
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Product-Coproduct Prographs and Triangulations of the Sphere
[article]

2022
*
arXiv
*
pre-print

In this paper, we explain how the classical Catalan families

arXiv:2202.05757v1
fatcat:ixcideilgzdgla2hb7hwlntheq
*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 ...##
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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. ...##
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Genus dependence of the number of (non-)orientable surface triangulations

2016
*
Physical Review D
*

We verify that the limit

doi:10.1103/physrevd.93.085018
fatcat:7j56lw4lxvbkjnzfdvhcqpl6oq
*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. ...##
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Cluster algebras of type D: pseudotriangulations approach
[article]

2015
*
arXiv
*
pre-print

Finally, we discuss applications

arXiv:1504.06377v1
fatcat:d2jw2n6xxrcuxpy4zw7wp5yoiu
*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 ...##
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Cluster Algebras of Type D: Pseudotriangulations Approach

2015
*
Electronic Journal of Combinatorics
*

Finally, we discuss applications

doi:10.37236/5282
fatcat:uv5c42zh5ncehkkjlnzab7ku3u
*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 ...##
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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 ...##
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On the mixing time of the flip walk on triangulations of the sphere
[article]

2018
*
arXiv
*
pre-print

A simple way

arXiv:1611.07324v2
fatcat:zda22xyhr5h6vkykm742vaf33q
*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. ...##
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Enumerating Constrained Non-crossing Minimally Rigid Frameworks

2007
*
Discrete & Computational Geometry
*

In particular, we obtain that the set

doi:10.1007/s00454-007-9026-x
fatcat:lrjbzrrcrrdahkopq7d7jf26du
*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*...##
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Short Encodings of Evolving Structures

1992
*
SIAM Journal on Discrete Mathematics
*

Similarly, it is shown that n(n1ogn) "

doi:10.1137/0405034
fatcat:geacnzwjh5amlnfkhggswigf3a
*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). ...##
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Pseudo-Triangulations - a Survey
[article]

2007
*
arXiv
*
pre-print

A pseudo-triangle is a simple polygon

arXiv:math/0612672v2
fatcat:adhyppd3wjbhnfmdky5dbxjxzu
*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. ...##
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Signed permutations and the four color theorem
[article]

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] ) ...

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

2021
*
arXiv
*
pre-print

Our starting point is a correspondence between counting perfect matchings in

arXiv:2005.13821v2
fatcat:vrqwkhpdnver7hhzzizl63pho4
*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 ...##
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Counting plane graphs: Perfect matchings, spanning cycles, and Kasteleyn's technique

2013
*
Journal of combinatorial theory. Series A
*

More specifically, we bound the ratio between the number

doi:10.1016/j.jcta.2013.01.002
fatcat:u62vgjo7o5azthn7qbwjybgixm
*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. ...
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