POINT VISIBILITY GRAPHS AND ${\mathcal O}$-CONVEX COVER
Bibliographic record
Abstract
A visibility relation can be viewed as a graph: the uncountable graph of a visibility relationship between points in a polygon P is called the point visibility graph (PVG) of P. In this paper we explore the use of perfect graphs to characterize tractable subproblems of visibility problems. Our main result is a characterization of which polygons are guaranteed to have weakly triangulated PVGs, under a generalized notion of visibility called [Formula: see text]-visibility. Let [Formula: see text] denote a set of line orientations. A connected point set P is called [Formula: see text]-convex if the intersection of P with any line with orientation in [Formula: see text] is connected. Two points in a polygon are said to be [Formula: see text]-visible if there is an [Formula: see text]-convex path between them inside the polygon. Let [Formula: see text] denote the set of orientations perpendicular to orientations in [Formula: see text]. Let [Formula: see text] be the set of orientations θ such that a "reflex" local maximum in the boundary of P exists with respect to θ. Our characterization of which polygons have weakly-triangulated PVGs is based on restricting the cardinality and span of [Formula: see text]. This characterization allows us to exhibit a class of polygons admitting a polynomial algorithm for [Formula: see text]-convex cover.
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How this classification was reachedexpand
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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".