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Simultaneous Diagonal Flips in Plane Triangulations

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Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Prosenjit Bose, Jurek Czyzowicz, Zhicheng Gao, Pat Morin, David R. Wood

Combinatorics · Submitted 21 Sep 2005

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This result is used to prove that for any two $n$-vertex triangulations, there exists a sequence of $O(\log n)$ simultaneous flips to transform one into the other.

how it is stated · asserted 1.00 read the record →

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arXiv IDmath/0509478 (version 2)
Pith IDCYHJS5GB, Pith's permanent record of this paper and its review
CitationDavid R. Wood, Jurek Czyzowicz, Pat Morin, Prosenjit Bose, Zhicheng Gao. "Simultaneous Diagonal Flips in Plane Triangulations." arXiv:math/0509478 (2005). https://pith.science/paper/CYHJS5GB
Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,171 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.35 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.08 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

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. It follows that every triangulation has a simultaneous flip into a Hamiltonian triangulation. This result is used to prove that for any two $n$-vertex triangulations, there exists a sequence of $O(\log n)$ simultaneous flips to transform one into the other. The total number of edges flipped in this sequence is O(n). The maximum size of a simultaneous flip is then studied. It is proved that every triangulation has a simultaneous flip of at least ${1/3}(n-2)$ edges. On the other hand, every simultaneous flip has at most $n-2$ edges, and there exist triangulations with a maximum simultaneous flip of ${6/7}(n-2)$ edges.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Simultaneous diagonal flips in plane triangulations are investigated. background 0.56 0.53 0.03 unclear 0.61
s1 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. result 0.99 0.94 0.47 asserted 1.00
s2 It follows that every triangulation has a simultaneous flip into a Hamiltonian triangulation. result 0.65 0.90 0.19 asserted 1.00
s3 This result is used to prove that for any two $n$-vertex triangulations, there exists a sequence of $O(\log n)$ simultaneous flips to transform one into the other. result 0.98 0.92 0.32 asserted 1.00
s4 The total number of edges flipped in this sequence is O(n). result 1.00 0.91 0.09 asserted 1.00
s5 The maximum size of a simultaneous flip is then studied. contribution 0.99 0.75 0.02 unclear 0.73
s6 It is proved that every triangulation has a simultaneous flip of at least ${1/3}(n-2)$ edges. result 1.00 0.93 0.15 asserted 1.00
s7 On the other hand, every simultaneous flip has at most $n-2$ edges, and there exist triangulations with a maximum simultaneous flip of ${6/7}(n-2)$ edges. result 1.00 0.90 0.08 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

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. It follows that every triangulation has a simultaneous flip into a Hamiltonian triangulation. This result is used to prove that for any two $n$-vertex triangulations, there exists a sequence of $O(\log n)$ simultaneous flips to transform one into the other. The total number of edges flipped in this sequence is O(n). The maximum size of a simultaneous flip is then studied. It is proved that every triangulation has a simultaneous flip of at least ${1/3}(n-2)$ edges. On the other hand, every simultaneous flip has at most $n-2$ edges, and there exist triangulations with a maximum simultaneous flip of ${6/7}(n-2)$ edges.

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