Bounds on the edge-length ratio of 2-outerplanar graphs
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Bibliographic record
Abstract
The edge-length ratio of a planar straight-line drawing Γ of a graph G is the largest ratio between the lengths of every pair of edges of Γ. If the ratio is measured by considering only pairs of edges that are incident to a common vertex, we talk about local edge-length ratio. The (local) edge-length ratio of a planar graph is the infimum over all (local) edge-length ratios of its planar straight-line drawings. It is known that the edge-length ratio of outerplanar graphs is upper bounded by a constant, while there exist graph families with non-constant outerplanarity that have non-constant lower bounds on their edge-length ratios. In this paper we prove an Ω ( n ) lower bound on the local edge-length ratio (and hence on the edge-length ratio) of the n -vertex 2-outerplanar graphs. We also prove a constant upper bound on the edge-length ratio of Halin graphs, pseudo-Halin graphs, and their generalizations.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| 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