Extending Certified Domination: Bondage Numbers in Generalized Petersen Graphs
Bibliographic record
Abstract
The paper extends the concept of bondage numbers to certified domination, introducing the certified bondage number of a graph. A certified dominating set R is a dominating set of a graph H, if every vertex in R has either zero or at least two neighbours in V\R, where V is the vertex set of H. The minimum cardinality of certified dominating set of H is the certified domination number of H denoted by γcer(H). The bondage number b(H) is defined to be the cardinality of least number of edges F ⊂ E(H) such that γ(H − F) > γ(H). Motivated by this parameter, we extended this concept on certified domination number and defined certified bondage number of a graph H, b+cer(H) [b−cer(H)] to be the cardinality of the least number of edges F ⊂ E(H) such that γcer(H−F) > γcer(H) [γcer(H−F) < γcer(H)] that is minimum number of edges to be removed to increase (or decrease) the certified domination number of H. In this paper, we establish the values of certified bondage number for generalised Petersen graphs P(n, k), where k = 1, 2, as well as for certain 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 | 0.001 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".