Product Structure of Graph Classes with Strongly Sublinear Separators
Bibliographic record
Abstract
We investigate the product structure of hereditary graph classes admitting strongly sublinear separators. We characterise such classes as subgraphs of the strong product of a star and a complete graph of strongly sublinear size. In a more precise result, we show that if any hereditary graph class 𝒢 admits O ( n 1 - ϵ ) separators, then for any fixed δ ∈ ( 0 , ϵ ) every n -vertex graph in 𝒢 is a subgraph of the strong product of a graph H with bounded tree-depth and a complete graph of size O ( n 1 - ϵ + δ ) . This result holds with δ = 0 if we allow H to have tree-depth O ( log log n ) . Moreover, using extensions of classical isoperimetric inequalties for grids graphs, we show the dependence on δ in our results and the above td ( H ) ∈ O ( log log n ) bound are both best possible. We prove that n -vertex graphs of bounded treewidth are subgraphs of the product of a graph with tree-depth t and a complete graph of size O ( n 1 / t ) , which is best possible. Finally, we investigate the conjecture that for any hereditary graph class 𝒢 that admits O ( n 1 - ϵ ) separators, every n -vertex graph in 𝒢 is a subgraph of the strong product of a graph H with bounded tree-width and a complete graph of size O ( n 1 - ϵ ) . We prove this for various classes 𝒢 of interest.
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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 | 0.001 | 0.003 |
| 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 | 0.002 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.010 | 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".