The class of (P7,C4,C5)‐free graphs: Decomposition, algorithms, and χ‐boundedness
Bibliographic record
Abstract
Abstract As usual, () denotes the path on vertices, and () denotes the cycle on vertices. For a family of graphs, we say that a graph is ‐free if no induced subgraph of is isomorphic to any graph in . We present a decomposition theorem for the class of ‐free graphs; in fact, we give a complete structural characterization of ‐free graphs that do not admit a clique‐cutset. We use this decomposition theorem to show that the class of ‐free graphs is ‐bounded by a linear function (more precisely, every ‐free graph satisfies ). We also use the decomposition theorem to construct an algorithm for the minimum coloring problem, an algorithm for the maximum weight stable set problem, and an algorithm for the maximum weight clique problem for this class, where denotes the number of vertices and the number of edges of the input graph.
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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".