Bibliographic record
Abstract
Given a graph G, a subset M of V(G) is a module of G if for each v∈V(G)∖M, v is adjacent to all the elements of M or adjacent to none of them. For instance, V(G), ∅ and {v} (v∈V(G)) are modules of G called trivial. Given a graph G, ωM(G) (respectively αM(G)) denotes the largest integer m such that there is a module M of G which is a clique (respectively a stable) set in G with |M|=m. A graph G is prime if |V(G)|≥4 and if all its modules are trivial. The prime bound of G is the smallest integer p(G) such that there is a prime graph H with V(H)⊇V(G), H[V(G)]=G and |V(H)∖V(G)|=p(G). We establish the following. For every graph G such that max(αM(G),ωM(G))≥2 and log2(max(αM(G),ωM(G))) is not an integer, p(G)=⌈log2(max(αM(G),ωM(G)))⌉. Then, we prove that for every graph G such that max(αM(G),ωM(G))=2k where k≥1, p(G)=k or k+1. Moreover p(G)=k+1 if and only if G or its complement admits exactly 2k isolated vertices. Lastly, we show that p(G)=1 for every non prime graph G such that |V(G)|≥4 and αM(G)=ωM(G)=1.
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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 | 0.002 | 0.014 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
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".