Bibliographic record
Abstract
Let G be a 2-connected graph with a toroidal rotation system given. An algorithm for constructing a straight line drawing with no crossings on a rectangular representation of the torus is presented. It is based on Read’s algorithm for constructing a planar layout of a 2-connected graph with a planar rotation system. It is proved that the method always works. The complexity of the algorithm is linear in the number of vertices of G. 1. Toroidal Graphs Let G be a toroidal graph, that is, one which can be drawn on the torus with no edge crossings. We require G to be a 2-connected graph, and we work only with 2-cell embeddings on the torus. The vertex and edge sets of G are V (G) and E(G), respectively. If u,v V (G), then u ! v means that u is adjacent to v (and so also v ! u). The reader is referred to Bondy and Murty [1] for other graph-theoretic terminology. G is represented by a rotation system, that is, the edges incident on each vertex v V (G) are cyclically ordered. This is sucient to determine the faces (2-cells) of the embedding. If G has n vertices, edges, and f faces, then Euler’s formula tells us that in a 2-cell embedding, n + f i = 0. Any rotation system which satisfies this formula is called a toroidal rotation system. We will find it useful to work with triangulations of the torus. In a triangulation, every face has degree 3, which gives us the further relations 2 = 6n = 3f. 1.1 Loops and Multiple Edges We will allow G to have loops and multiple edges. This is necessary, since the duals of graphs we are interested in will often have loops or multiple edges. However, if vv is a loop, we require that the cycle vv be an essential cycle of the embedding, that is, if the torus is cut along the cycle vv, the * This work was supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.015 | 0.003 |
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 source (direct Gemma or distilled Codex), 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".