The edge surviving rate of Halin graphs
Bibliographic record
Abstract
<p>Let <span class="math inline">\(k\ge 1\)</span> be an integer. Let <span class="math inline">\(G=(V,E)\)</span> be a connected graph with <span class="math inline">\(n\)</span> vertices and <span class="math inline">\(m\)</span> edges. Suppose fires break out at two adjacent vertices. In each round, a firefighter can protect <span class="math inline">\(k\)</span> vertices, and then the fires spread to all unprotected neighbors. For <span class="math inline">\(uv\in E(G)\)</span>, let <span class="math inline">\(sn_{k}(uv)\)</span> denote the maximum number of vertices the firefighter can save when fires break out at the ends of <span class="math inline">\(uv\)</span>. The <span class="math inline">\(k\)</span>-edge surviving rate <span class="math inline">\(\rho&#39;_k(G)\)</span> of <span class="math inline">\(G\)</span> is defined as the average proportion of vertices saved when the starting vertices of the fires are chosen uniformly at random over all eages, i.e., <span class="math inline">\(\rho&#39;_k(G)=\sum\limits_{uv\in E(G)}sn_{k}(uv)/nm\)</span>. In particular, we write <span class="math inline">\(\rho&#39;(G)=\rho&#39;_1(G)\)</span>. For a given class of graphs <span class="math inline">\(\mathcal{G}\)</span> and a constant <span class="math inline">\(\varepsilon>0\)</span>, we seek the minimum value <span class="math inline">\(k\)</span> such that <span class="math inline">\(\rho&#39;_k(G)>\varepsilon\)</span> for all <span class="math inline">\(G\in \mathcal{G}\)</span>. In this paper, we prove that for Halin graphs, this minimum value is exactly 1. Specifically, every Halin graph <span class="math inline">\(G\)</span> satisfies <span class="math inline">\(\rho&#39;(G)> 1/12\)</span>.</p>
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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 | 0.018 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.012 | 0.002 |
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".