Bibliographic record
Abstract
A perfect dominating set in a graph G = (V, E) is a subset S ⊆ V such that each vertex in V \ S has exactly one neighbor in S . A perfect coalition in G consists of two disjoint sets of vertices V 1 and V 2 such that i) neither V 1 nor V 2 is a dominating set, ii) each vertex in V(G) \ V 1 has at most one neighbor in V 1 and each vertex in V(G) \ V 2 has at most one neighbor in V 2 , and iii) V 1 ∪ V 2 is a perfect dominating set. A perfect coalition partition (abbreviated prc -partition) in a graph G is a vertex partition π = {V 1 , V 2 , …, V k } such that for each set V i of π , either V i is a singleton dominating set or there exists a set V j ∈ π that forms a perfect coalition with V i . In this paper, we initiate the study of perfect coalition partitions in graphs. We obtain a bound on the number of perfect coalitions involving each member of a perfect coalition partition, in terms of maximum degree. The perfect coalition of some special graphs are investigated. Graphs with minimum degree one, triangle-free graphs and trees with large perfect coalition numbers are investigated.
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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 | 0.002 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.005 |
| Science and technology studies | 0.003 | 0.005 |
| Scholarly communication | 0.004 | 0.007 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".