Total Roman domination edge-critical graphs
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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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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it