Simultaneous diagonal flips in plane triangulations
Bibliographic record
Abstract
Abstract Simultaneous diagonal flips in plane triangulations are investigated. It is proved that every triangulation with n ≥ 6 vertices has a simultaneous flip into a 4‐connected triangulation, and that the set of edges to be flipped can be computed in $\cal O$ (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 $\cal O$ (logn) simultaneous flips to transform one into the other. Moreover, Ω(log n) simultaneous flips are needed for some pairs of triangulations. The total number of edges flipped in this sequence is $\cal O$ (n). The maximum size of a simultaneous flip is then studied. It is proved that every triangulation has a simultaneous flip of at least ${{1}\over{3}}({n}-{2})$ edges. On the other hand, every simultaneous flip has at most n − 2 edges, and there exist triangulations with a maximum simultaneous flip of ${{6}\over{7}}({n}-{2})$ edges. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 307–330, 2007
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.010 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".