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Simultaneous diagonal flips in plane triangulations

2006
*
Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06
*

*Simultaneous*

*diagonal*

*flips*

*in*

*plane*

*triangulations*are investigated. ... Thanks to the referees for pointing out an error

*in*a preliminary version of the paper. ... This operation is called a

*simultaneous*(

*diagonal*)

*flip*. ...

##
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Simultaneous diagonal flips in plane triangulations

2007
*
Journal of Graph Theory
*

*Simultaneous*

*diagonal*

*flips*

*in*

*plane*

*triangulations*are investigated. ... It is proved that every n-vertex

*triangulation*with at least six vertices has a

*simultaneous*

*flip*into a 4-connected

*triangulation*, and that it can be computed

*in*O(n) time. ... Thanks to the referees for pointing out an error

*in*a preliminary version of the paper. ...

##
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Diagonal flips in Hamiltonian triangulations on the projective plane

2005
*
Discrete Mathematics
*

*In*this paper, we shall prove that any two

*triangulations*on the projective

*plane*with n vertices can be transformed into each other by at most 8n − 26

*diagonal*

*flips*, up to isotopy. ... To prove it, we focus on

*triangulations*on the projective

*plane*with contractible Hamilton cycles. ... Observe that the degree of x and y decrease

*simultaneously*by one

*diagonal*

*flip*, only if this

*diagonal*

*flip*is applied to the edge xy. ...

##
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SIMULTANEOUS EDGE FLIPPING IN TRIANGULATIONS

2003
*
International journal of computational geometry and applications
*

We generalize the operation of

doi:10.1142/s0218195903001098
fatcat:aya24vpgyjdrlndb2awxefsypi
*flipping*an edge*in*a*triangulation*to that of*flipping*several edges*simultaneously*. ... Our main result is an optimal upper bound on the number of*simultaneous**flips*that are needed to transform a*triangulation*into another. ...*In*a previous paper [14] , the authors studied several questions about*flips**in**triangulations*, mainly the question of how many*flips*are needed to transform a*triangulation*of a*plane*point set (or of ...##
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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 ... We overview both the combinatorial perspective (where only a combinatorial embedding of the graph is specified) and the geometric perspective (where the graph is embedded*in*the*plane*, vertices are points ...*Diagonal**flips**in*the near-*triangulation*correspond isomorphically to rotations*in*the tree. ...##
###
Flipping edge-labelled triangulations

2018
*
Computational geometry
*

*Flips*

*in*

*triangulations*have received a lot of attention over the past decades. However, the problem of tracking where particular edges go during the

*flipping*process has not been addressed. ... When

*simultaneous*

*flips*are allowed, the upper bound for convex polygons decreases to O(log 2 n), although we no longer have a matching lower bound. ... an intermediate form between unlabelled combinatorial

*triangulations*and

*triangulations*of a set of points

*in*the

*plane*. ...

##
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Flipping Edge-Labelled Triangulations
[article]

2016
*
arXiv
*
pre-print

*Flips*

*in*

*triangulations*have received a lot of attention over the past decades. However, the problem of tracking where particular edges go during the

*flipping*process has not been addressed. ... When

*simultaneous*

*flips*are allowed, the upper bound for convex polygons decreases to O(^2 n), although we no longer have a matching lower bound. ... [26] as an intermediate form between unlabelled combinatorial

*triangulations*and

*triangulations*of a set of points

*in*the

*plane*. ...

##
###
Computing the Flip Distance Between Triangulations

2017
*
Discrete & Computational Geometry
*

Let T be a

doi:10.1007/s00454-017-9867-x
fatcat:hkjet6cuyrcz5ltmnn2i6jjwie
*triangulation*of a set P of n points*in*the*plane*, and let e be an edge shared by two triangles*in*T such that the quadrilateral Q formed by these two triangles is convex. ... A*flip*of e is the operation of replacing e by the other*diagonal*of Q to obtain a new*triangulation*of P from T . ... Introduction Let P be a set of n points*in*the*plane*. ...##
###
Counting Plane Graphs: Flippability and its Applications
[article]

2011
*
arXiv
*
pre-print

We generalize the notions of flippable and

arXiv:1012.0591v2
fatcat:7entxgoar5f7nbiwwbpnyoasuy
*simultaneously*flippable edges*in*a*triangulation*of a set S of points*in*the*plane*to so-called pseudo-*simultaneously*flippable edges. ... Specifically, let tr(N) denote the maximum number of*triangulations*on a set of N points*in*the*plane*. ... Lemma 1.1 [19] For any*triangulation*T over a set of N points*in*the*plane*,*flip*(T ) ≥ N/2 − 2. ...##
###
Counting Plane Graphs: Flippability and Its Applications
[chapter]

2012
*
Thirty Essays on Geometric Graph Theory
*

We generalize the notions of flippable and

doi:10.1007/978-1-4614-0110-0_16
fatcat:m52k5642lfff7ak7gwh3trmmqa
*simultaneously*-flippable edges*in*a*triangulation*of a set S of points*in*the*plane*, into so called pseudo*simultaneously*-flippable edges. ... More specifically, denoting by tr(N ) < 30 N the maximum possible number of*triangulations*of a set of N points*in*the*plane*, we show that every set of N points*in*the*plane*can have at most 6.9283 N · ... [10] , as the*simultaneous*-flippability lemma. Lemma 1.2 For any*triangulation*T over a set of N points*in*the*plane*,*flip*s (T ) ≥*flip*(T ) 3 .*In*particular,*flip*s (T ) ≥ N/6 − 2/3. Galtier et al ...##
###
Counting Plane Graphs: Flippability and Its Applications
[chapter]

2011
*
Lecture Notes in Computer Science
*

We generalize the notions of flippable and

doi:10.1007/978-3-642-22300-6_44
fatcat:vudokcakg5gp5fljxubjtofudy
*simultaneously*-flippable edges*in*a*triangulation*of a set S of points*in*the*plane*, into so called pseudo*simultaneously*-flippable edges. ... More specifically, denoting by tr(N ) < 30 N the maximum possible number of*triangulations*of a set of N points*in*the*plane*, we show that every set of N points*in*the*plane*can have at most 6.9283 N · ... [10] , as the*simultaneous*-flippability lemma. Lemma 1.2 For any*triangulation*T over a set of N points*in*the*plane*,*flip*s (T ) ≥*flip*(T ) 3 .*In*particular,*flip*s (T ) ≥ N/6 − 2/3. Galtier et al ...##
###
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. ... We use several facts about*diagonal**flips**in**triangulations*of a FACT 1. ...##
###
A Bound on the Edge-Flipping Distance between Triangulations (Revisiting the Proof)
[article]

2021
*
arXiv
*
pre-print

We revisit here a fundamental result on planar

arXiv:2106.14408v1
fatcat:4tuvlxtwe5ctrf6i2v2idiwk2u
*triangulations*, namely that the*flip*distance between two*triangulations*is upper-bounded by the number of proper intersections between their straight-segment ... We provide a complete and detailed proof of this result*in*a slightly generalised setting using a case-based analysis that fills several gaps left by previous proofs of the result. ... Edge-*flipping*distance and intersections: an upper-bound. We consider*triangulations*of a fixed finite set of points S*in*the*plane*. ...##
###
Computing Motorcycle Graphs Based on Kinetic Triangulations

2012
*
Canadian Conference on Computational Geometry
*

*In*order to reduce the number of

*flip*events we investigate the use of Steiner

*triangulations*. ... Our algorithm applies kinetic

*triangulations*inside of the convex hull of the input, while a

*plane*sweep is used outside of it. ... If the event associated with T 2 is handled first then we

*flip*the

*diagonal*(b, d) to the

*diagonal*(a, c), resulting

*in*the triangles ∆(a, b, c) and ∆(a, c, d). ...

##
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Counting Plane Graphs: Perfect Matchings, Spanning Cycles, and Kasteleyn's Technique
[article]

2011
*
arXiv
*
pre-print

the

arXiv:1109.5596v1
fatcat:pbtrxyjgajh5tcsxmfdu3esjgm
*plane*. ... These imply a new upper bound of O(54.543^N) on the number of crossing-free straight-edge spanning cycles that can be embedded over any specific set of N points*in*the*plane*(improving upon the previous ... Ps-flippable edges*In*[9] , we present the concept of pseudo*simultaneously*flippable edges (or ps-flippable edges, for short). ...
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