Bibliographic record
Abstract
In their Graph Minors Project, Robertson and Seymour proved that, in any infinite set of graphs, one is a minor of another. In particular, if S is a surface, the set of minor-minimal graphs that are not embeddable in S is finite. Two central results of the Graph Minors Project are: if the graphs in the infinite set have bounded tree-width, then one is a minor of the other; graphs with large tree-width have large grids as minors. We present the ‘simple’ proofs of these two facts, and adapt an argument of Thomassen that shows how to apply them to prove the finiteness of the set of minor-minimal non-S-embeddable graphs. Introduction This chapter is a self-contained introduction to graph minors. It contains a complete proof of the generalization of Kuratowski's theorem to higher surfaces; more importantly, it is a major step in understanding the whole Graph Minors Project of Robertson and Seymour. The only background needed is some familiarity with connectivity issues (essentially variations of Menger's theorem and a willingness to view cutsets from different perspectives). Our experience with the arguments presented here is that we need to be able to focus on both the big picture and on the details. There are many small points that require their own little arguments and we have attempted to provide these in sufficient detail to make it easier not to lose sight of the big picture.
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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 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".