Product Structure Extension of the Alon–Seymour–Thomas Theorem
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Bibliographic record
Abstract
Alon, Seymour, and Thomas [J. Amer. Math. Soc., 3 (1990), pp. 801-808] proved that every n-vertex graph excluding Kt as a minor has treewidth less than t3/2 \\surdn. Illingworth, Scott, and Wood [Product Structure of Graphs with an Excluded Minor, preprint, arXiv:2104.06627, 2022] recently refined this result by showing that every such graph is a subgraph of some graph with treewidth t - 2, where each vertex is blown up by a complete graph of order \\scrO(\\surdtn). Solving an open problem of Illingworth, Scott, and Wood [2022], we prove that the treewidth bound can be reduced to 4 while keeping blowups of order \\scrOt(\\surdn). As an extension of the Lipton-Tarjan theorem, in the case of planar graphs, we show that the treewidth can be further reduced to 2, which is best possible. We generalize this result for K3,t-minor-free graphs, with blowups of order \\scrO(t\\surdn). This setting includes graphs embeddable on any fixed surface.
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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 | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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