On the geometric thickness of 2-degenerate graphs
Bibliographic record
Abstract
A graph is $2$-degenerate if every subgraph contains a vertex of degree at most $2$. We show that every $2$-degenerate graph can be drawn with straight lines such that the drawing decomposes into $4$ plane forests. Therefore, the geometric arboricity, and hence the geometric thickness, of $2$-degenerate graphs is at most $4$. On the other hand, we show that there are $2$-degenerate graphs that do not admit any straight-line drawing with a decomposition of the edge set into $2$ plane graphs. That is, there are $2$-degenerate graphs with geometric thickness, and hence geometric arboricity, at least $3$. This answers two questions posed by Eppstein [Separating thickness from geometric thickness. In: Towards a Theory of Geometric Graphs, volume 342 of Contemp. Math., AMS, 2004].
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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 | 0.004 | 0.008 |
| 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 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".