Boolean sum of graphs and reconstruction up to complementation
Bibliographic record
Abstract
Let V be a set of cardinality v (possibly infinite). Two graphs G and with vertex set V are isomorphic up to complementation if is isomorphic to G or to the complement of G . Let k be a non-negative integer. The graphs G and are k -hypomorphic up to complementation if for every k -element subset K of V , the induced subgraphs and are isomorphic up to complementation. A graph G is k -reconstructible up to complementation if every graph which is k -hypomorphic to G up to complementation is in fact isomorphic to G up to complementation. We prove that a graph G has this property provided that . Moreover, under these conditions, if or , then G and are the only graphs k -hypomorphic to G up to complementation. A description of pairs of graphs with the same 3-homogeneous subsets is a key ingredient in our proof.
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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 | 0.012 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.005 | 0.008 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.007 | 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".