Determination of the prime bound of a graph
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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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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.000 |
| 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