Encoding and enumerating acyclic orientations of graphs
Bibliographic record
Abstract
<p>In this work we study the acyclic orientations of graphs. We obtain an encoding of the acyclic orientations of the complete <span class="math inline">\(p\)</span>-partite graph with size of its parts <span class="math inline">\(n_1,n_2,\ldots,n_p\)</span> via a vector with <span class="math inline">\(p\)</span> symbols and length <span class="math inline">\(n=n_1+n_2+\ldots+n_p\)</span> when the parts are fixed but not the vertices in each part. We also give a recursive way to construct all acyclic orientations of a complete multipartite graph, this construction can be done by computer easily in order <span class="math inline">\(\mathcal{O}(n)\)</span>. Furthermore, we obtain a closed formula for non-isomorphic acyclic orientations of both the complete multipartite graphs and the complete multipartite graphs with a directed spanning tree. Moreover, we obtain a closed formula for the number of acyclic orientations of a complete multipartite graph <span class="math inline">\(K_{n_1,\ldots,n_p}\)</span> with labelled vertices. Finally, we obtain a way encode all acyclic orientations of an arbitrary graph as a permutation code. Using the codification mentioned above we obtain sharp upper and lower bounds of the number of acyclic orientations of a graph.</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 | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 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; 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".