Drawing planar graphs with prescribed face areas
Why this work is in the frame
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Bibliographic record
Abstract
We study drawings of planar graphs where every inner face has a prescribed area. A plane graph is 'area-universal' if for every area assignment on the inner faces, there exists a straight-line drawing realizing the assigned areas. The only non-area-universal graph known so far is the octahedron graph. We give a simple counting argument that allows to prove non-area-universality for a large class of triangulations, namely Eulerian triangulations. Adding some geometric arguments, the concept allows to prove non-area universality for other graphs including the icosahedron. Relaxing the straight-line property by allowing the edges to bend, we show that one bend per edge is sufficient to realize any face area assignment of every plane graph. For plane bipartite graphs, it even suffices that half of the edges have a bend.
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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 | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.002 |
| 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