Hypergraph burning, matchings, and zero forcing
Bibliographic record
Abstract
Lazy burning is a recently introduced variation of burning where only one set of vertices is chosen to burn during the first round. In hypergraphs, lazy burning spreads when all but one vertex in a hyperedge is burned. The lazy burning number is the minimum number of initially burned vertices that eventually burn all vertices. We give several equivalent characterizations of lazy burning on hypergraphs using matchings and zero forcing, and then apply these to establish new bounds and complexity results. We prove that the lazy burning number of a hypergraph H equals its order minus the maximum cardinality of a certain matching on its incidence graph. Using this characterization, we give a formula for the lazy burning number of a dual hypergraph and give new bounds on the lazy burning number based on various hypergraph parameters. We show that the lazy burning number of a hypergraph may be characterized by a maximal subhypergraph that results from iteratively deleting vertices in singleton hyperedges. We prove that lazy burning on a hypergraph is equivalent to zero forcing on its incidence graph and show an equivalence between skew zero forcing on a graph and lazy burning on its neighborhood hypergraph. As a result, we show that the decision problem of computing the lazy burning number of a hypergraph is NP-complete, which solves an open problem in [12] . By applying the results found for lazy burning, we show that the decision problem of computing the skew zero forcing number for bipartite graphs is NP-complete. We finish with open problems.
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How this classification was reachedexpand
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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".