Computational Analysis of Certified Reinforcement Numbers Across Specialized Graph Classes
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Bibliographic record
Abstract
A certified dominating set D of a graph G is a dominating set in which every vertex in D must have either no neighbors or at least two neighbors in V\D, where V denotes the set of all vertices in G. A certified domination number of G represented by γcer(G) is defined as the smallest size of such a certified dominating set of G. The reinforcement number r(G) is defined to be the cardinality of minimum number of edges F ⊂ E(Gˉ) such that γ(G + F) < γ(G), broadened this parameter to encompass certified domination and we define certified reinforcement number of a graph G, rcer(G) to be the cardinality of the minimum number of edges F ⊂ E(¯G) such that γcer(G + F) < γcer(G) that is minimum number of edges to be added to decrease the certified domination number of G at least by one. In this paper, we characterize the graph G for which rcer(G) = 1 and determine the values of certified reinforcement number for various classes of graphs.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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 it