Examining Captive and Inverse Captive Domination in Selected Graphs and Their Complements
Bibliographic record
Abstract
The aim of this paper is to present new properties of captive domination and determine the number of some graphs.The proper subset of the vertices of a graph G is a captive dominating set if it is a total dominating set and each vertex in this set dominates at least one vertex which does not belong to the dominating set.The domination number 𝛾(𝐺) is the minimum cardinality of a dominating set D of G.If V-D contains a dominating set, then this set is called an inverse set of D in G.The symbol 𝛾 -1 (𝐺) represents the minimum cardinality over all inverse dominating set of G. Some graphs which determine the captive domination number such as a ladder graph, corona graph of two paths, lollipop graph, barbell graph, corona graph of a cycle of order n, and null graph of order p and helm graph.For all these graphs and complements the captive domination and inverse captive domination are calculated.
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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 | 0.007 |
| 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 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".