Dominance Parameters in Prism Graphs: A Comparative Study of Minimum Dominating Sets
Bibliographic record
Abstract
Let G=(V, E) be a graph.A dominating set S of graph G is defined as a set of vertices such that every vertex in V\S is adjacent to at least one vertex in S. The domination number of graph G, denoted as γ(G), corresponds to the size of the smallest dominating set within G.In other words, γ(G) represents the number of vertices required in the minimum dominating set to cover all other vertices in the graph G.In the graph G, our objective is to position a protector at each vertex within a subset S of V, ensuring that S forms a dominating set, effectively covering all other vertices in G.Moreover, in the event that a protector positioned at vertex 𝑣 needs to move along an edge to protect an unguarded vertex u, the arrangement of protectors should maintain the property of forming a dominating set for the graph.In other words, the movement of protectors should maintain the property of domination within the graph, ensuring efficient coverage and defense across the network.The bare minimum of security guards is necessary to protect all vertices in the graphs.In this article, we find the bounds for domination, independent domination number (IDN), connected domination number(CDN), total domination number(TDN), and the secure domination number(SDN) denoted byγ(A n ), γ i (A n ), γ c (A n ), γ t and γ s (A n ) respectively for the antiprism graph, where A n denoted the 4 -regular graph with girth 3. We further establish that the TDN is greater than or equal to the SDN of the antiprism graph for 𝑛 ≥ 3.
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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 | 0.005 | 0.023 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.004 | 0.004 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| 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".