Computational Analysis of Certified Reinforcement Numbers Across Specialized Graph Classes
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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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 | 0.031 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".