A note on the lower bound of edge guards of polyhedral terrains
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Bibliographic record
Abstract
Bose et al. [P. Bose et al., Guarding polyhedral terrains, Comput. Geom., Theory Appl. 6 (1997), pp. 173–185.] proved that ⌊ (4n−4)/13 ⌋ edge guards are sometimes necessary to guard an n-vertex polyhedral terrain. Subsequently, Kaučič et al. [B. Kaučič, B. Žalik, and F. Novak, On the lower bound of edge guards of polyhedral terrains, Int. J. Comput. Math. 80 (2003), pp. 811–814.] claimed to find an inconsistency in the proof and used Bose et al. ’s proof technique to prove a weaker lower bound of ⌊ (2n−4)/7 ⌋ edge-guards. They declared that a proof of the original lower bound of ⌊ (4n−4)/13 ⌋ remains an open issue. The purpose of this note is simply to point out that the issue is not open and that Bose et al. ’s original proof is correct. We present the original proof of ⌊ (4n−4)/13 ⌋ at a level of detail to hopefully remove any misunderstanding of the result.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it