Bibliographic record
Abstract
<p>For a graph <span class="math inline">\(G\)</span>, an Italian dominating function (IDF) is a function <span class="math inline">\(f: V(G) \rightarrow \{0,1,2\}\)</span> such that all vertices labeled with 0 must have at least two neighbors assigned the label 1 or at least one neighbor assigned the label 2. The weight of <span class="math inline">\(f\)</span>, denoted by <span class="math inline">\(w(f)\)</span>, is calculated by summing all the labels assigned by the function. Let <span class="math inline">\(f\)</span> be an IDF on <span class="math inline">\(G\)</span> with a minimum weight, denoted as <span class="math inline">\(\gamma_I(G)\)</span>. If <span class="math inline">\(S\)</span> is the set of vertices where <span class="math inline">\(f(v) > 0\)</span>, then an Inverse Italian Dominating Function (IIDF) <span class="math inline">\(f&#39;\)</span> is defined as an IDF on <span class="math inline">\(G\)</span> such that <span class="math inline">\(f&#39;(v) = 0\)</span> for all <span class="math inline">\(v \in S\)</span>. The notation <span class="math inline">\(\gamma_{iI}(G)\)</span> represents the Inverse Italian Domination Number of the graph <span class="math inline">\(G\)</span>, which is the minimum weight among all IIDFs on <span class="math inline">\(G\)</span>. In this paper, we find <span class="math inline">\(\gamma_{iI}(G)\)</span> of graphs and characterize the graphs for which <span class="math inline">\(\gamma_I(G) = 2\)</span> and <span class="math inline">\(3\)</span>, as well as those with <span class="math inline">\(\gamma_{iI}(G) = 2\)</span> and <span class="math inline">\(3\)</span>. Additionally, we provide a characterization of trees and graphs that achieve the largest possible <span class="math inline">\(\gamma_{iI}(G)\)</span>.</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 | 0.001 | 0.000 |
| 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 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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; both teacher heads agree on what is shown here.
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".