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Simultaneous diagonal flips in plane triangulations

Prosenjit Bose, Jurek Czyzowicz, Zhicheng Gao, Pat Morin, David R. Wood
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.  ... 
doi:10.1145/1109557.1109582 fatcat:xxsgxyohcrbi3ab3epo5llrhea

Simultaneous diagonal flips in plane triangulations

Prosenjit Bose, Jurek Czyzowicz, Zhicheng Gao, Pat Morin, David R. Wood
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.  ... 
doi:10.1002/jgt.20214 fatcat:67ykyefwrrbfzieq3qt2wcwgsq

Diagonal flips in Hamiltonian triangulations on the projective plane

Ryuichi Mori, Atsuhiro Nakamoto
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.  ... 
doi:10.1016/j.disc.2004.12.021 fatcat:bpglpvj2vbbgxamcmb7etpqlmq

SIMULTANEOUS EDGE FLIPPING IN TRIANGULATIONS

JERÔME GALTIER, FERRAN HURTADO, MARC NOY, STÉPHANE PÉRENNES, JORGE URRUTIA
2003 International journal of computational geometry and applications  
We generalize the operation of 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  ... 
doi:10.1142/s0218195903001098 fatcat:aya24vpgyjdrlndb2awxefsypi

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  ...  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.  ... 
doi:10.1016/j.comgeo.2008.04.001 fatcat:fxq5ql3imzftpoh5naidvy5e3q

Flipping edge-labelled triangulations

Prosenjit Bose, Anna Lubiw, Vinayak Pathak, Sander Verdonschot
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.  ... 
doi:10.1016/j.comgeo.2017.06.005 fatcat:txhxzy2al5fx3ppqkx77iqdcki

Flipping Edge-Labelled Triangulations [article]

Prosenjit Bose, Anna Lubiw, Vinayak Pathak, Sander Verdonschot
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.  ... 
arXiv:1310.1166v2 fatcat:nheg34gqhvgnhmx5l7mall4t6a

Computing the Flip Distance Between Triangulations

Iyad Kanj, Eric Sedgwick, Ge Xia
2017 Discrete & Computational Geometry  
Let T be a 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.  ... 
doi:10.1007/s00454-017-9867-x fatcat:hkjet6cuyrcz5ltmnn2i6jjwie

Counting Plane Graphs: Flippability and its Applications [article]

Michael Hoffmann, Micha Sharir, Adam Sheffer, Csaba D. Tóth, Emo Welzl
2011 arXiv   pre-print
We generalize the notions of flippable and 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.  ... 
arXiv:1012.0591v2 fatcat:7entxgoar5f7nbiwwbpnyoasuy

Counting Plane Graphs: Flippability and Its Applications [chapter]

Michael Hoffmann, André Schulz, Micha Sharir, Adam Sheffer, Csaba D. Tóth, Emo Welzl
2012 Thirty Essays on Geometric Graph Theory  
We generalize the notions of flippable and 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  ... 
doi:10.1007/978-1-4614-0110-0_16 fatcat:m52k5642lfff7ak7gwh3trmmqa

Counting Plane Graphs: Flippability and Its Applications [chapter]

Michael Hoffmann, Micha Sharir, Adam Sheffer, Csaba D. Tóth, Emo Welzl
2011 Lecture Notes in Computer Science  
We generalize the notions of flippable and 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  ... 
doi:10.1007/978-3-642-22300-6_44 fatcat:vudokcakg5gp5fljxubjtofudy

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.  ...  We use several facts about diagonal flips in triangulations of a FACT 1.  ... 
doi:10.1137/0405034 fatcat:geacnzwjh5amlnfkhggswigf3a

A Bound on the Edge-Flipping Distance between Triangulations (Revisiting the Proof) [article]

Thomas Dagès, Alfred M. Bruckstein
2021 arXiv   pre-print
We revisit here a fundamental result on planar 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.  ... 
arXiv:2106.14408v1 fatcat:4tuvlxtwe5ctrf6i2v2idiwk2u

Computing Motorcycle Graphs Based on Kinetic Triangulations

Willi Mann, Martin Held, Stefan Huber
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).  ... 
dblp:conf/cccg/MannHH12 fatcat:jii3tf3dkrb5rhagxf6hl4ulyu

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

Micha Sharir, Adam Sheffer, Emo Welzl
2011 arXiv   pre-print
the 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).  ... 
arXiv:1109.5596v1 fatcat:pbtrxyjgajh5tcsxmfdu3esjgm
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