Bibliographic record
Abstract
<p>A <span class="math inline">\(\{2\}\)</span>-dominating function (<span class="math inline">\(\{2 \}\)</span>DF) on a graph <span class="math inline">\(G=(V(G),E(G))\)</span> is a function <span class="math inline">\(f : V(G) \rightarrow \{0,1,2 \}\)</span> such that <span class="math inline">\(f(N[v]) \geq 2\)</span> for every <span class="math inline">\(v \in V(G)\)</span>, where <span class="math inline">\(N[v]\)</span> is the closed neighourhood of <span class="math inline">\(v\)</span>. The <span class="math inline">\(\{2\}\)</span>-domination number of <span class="math inline">\(G\)</span> is the minimum weight <span class="math inline">\(\omega(f) = \sum\limits_{v \in V(G)} f(v)\)</span> among all <span class="math inline">\(\{2 \}\)</span>-dominating functions on <span class="math inline">\(G\)</span>. In this article, we prove that if <span class="math inline">\(G\)</span> and <span class="math inline">\(H\)</span> are graphs with no isolated vertex, then for any vertex <span class="math inline">\(v \in V(H)\)</span> there are six closed formulas for the <span class="math inline">\(\{2\}\)</span>-domination number of the rooted product graph <span class="math inline">\(G \circ_v H\)</span>. We also characterize the graph <span class="math inline">\(G\)</span> and <span class="math inline">\(H\)</span> that satisfy each of these formulas.</p>
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How this classification was reachedexpand
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 | 0.005 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".