Bibliographic record
Abstract
We improve the generalized upper bound for the k-tuple domination number given in [A. Gagarin and V.E. Zverovich, A generalized upper bound for the k-tuple domination number, Discrete Math. 308 no. 5–6 (2008), 880–885]. Precisely, we show that for any graph G, when k = 3, or k = 4 and d ≤ 3.2, γ×k(G) ≤ ln(δ−k + 2) + ln (k − 2)d + ∑k−2 m=2 δ − k + 2 and, when k = 4 and d> 3.2, or k ≥ 5, γ×k(G) ≤ ln(δ − k + 2) + ln ( ∑k−2 (k−m) m=0 4 δ − k + 2 (k−m) 4min{m, k−2−m} dm + ) dk−1 dm min{m, k−2−m} + dk−1 + 1 n, + 1 n, where γ×k(G) is the k-tuple domination number, δ is the minimum degree, d is the average degree, and dm is the m-degree of G. Moreover, when k ≥ 5, the latter bound can be improved to γ×k(G) ≤ ln(δ − k + 2) + ln ( ∑k−2 (k−m) dm m=0 + P(k−2,m) ) dk−1 + 1 n, δ − k + 2 where the coefficient P(t, m) = P(t, t) = 1, with t = k − 2. t t m m (t−m) t−m for t> m> 0, P(t, 0) = 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 | 0.003 | 0.016 |
| Meta-epidemiology (narrow) | 0.004 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.003 |
| Bibliometrics | 0.004 | 0.005 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.004 | 0.010 |
| Open science | 0.005 | 0.006 |
| Research integrity | 0.002 | 0.007 |
| Insufficient payload (model declined to judge) | 0.011 | 0.004 |
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".