Total Roman domination edge-critical graphs
Bibliographic record
Abstract
A total Roman dominating function on a graph [math] is a function [math] such that every vertex [math] with [math] is adjacent to some vertex [math] with [math] , and the subgraph of [math] induced by the set of all vertices [math] such that [math] has no isolated vertices. The weight of [math] is [math] . The total Roman domination number [math] is the minimum weight of a total Roman dominating function on [math] . A graph [math] is [math] - [math] -edge-critical if [math] for every edge [math] , and [math] - [math] -edge-supercritical if it is [math] - [math] -edge-critical and [math] for every edge [math] . We present some basic results on [math] -edge-critical graphs and characterize certain classes of [math] -edge-critical graphs. In addition, we show that, when [math] is small, there is a connection between [math] - [math] -edge-critical graphs and graphs which are critical with respect to the domination and total domination numbers.
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| 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".