Structure and Complexity of 2-Intersection Graphs of 3-Hypergraphs
Bibliographic record
Abstract
Abstract Given a 3-uniform hypergraph H having a set V of vertices, and a set of hyperedges $$T\subset \mathcal {P}(V)$$ T ⊂ P ( V ) , whose elements have cardinality three each, a null labelling is an assignment of $$\pm 1$$ ± 1 to the hyperedges such that each vertex belongs to the same number of hyperedges labelled $$+1$$ + 1 and $$-1$$ - 1 . A sufficient condition for the existence of a null labelling of H (proved in Di Marco et al. Lect Notes Comput Sci 12757:282–294, 2021) is a Hamiltonian cycle in its 2-intersection graph. The notion of 2-intersection graph generalizes that of intersection graph of an (hyper)graph and extends its effectiveness. The present study first shows that this sufficient condition for the existence of a null labelling in H can not be weakened by requiring only the connectedness of the 2-intersection graph. Then some interesting properties related to their clique configurations are proved. Finally, the main result is proved, the NP-completeness of this characterization and, as a consequence, of the construction of the related 3-hypergraphs.
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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 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.004 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.008 | 0.010 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.013 | 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".