On multiset dimension of cylindrical graphs
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Bibliographic record
Abstract
Let <span class="math inline"><em>G</em> = (<em>V</em>, <em>E</em>)</span> be a simple connected graph and <span class="math inline"><em>W</em> ⊆ <em>V</em>.</span> For <span class="math inline"><em>v</em> ∈ <em>V</em>,</span> the representation multiset or m-code of <span class="math inline"><em>v</em></span> is the multiset <span class="math inline"><em>r</em><sub><em>m</em></sub>(<em>v</em>) = {<em>d</em>(<em>v</em>, <em>w</em>) ∣ <em>w</em> ∈ <em>W</em>}</span>. If no two vertices in <span class="math inline"><em>G</em></span> have equal m-codes, then <span class="math inline"><em>W</em></span> is called an m-resolving set of <span class="math inline"><em>G</em></span>. The multiset dimension <span class="math inline">md(<em>G</em>)</span> of <span class="math inline"><em>G</em></span> is the minimum possible cardinality of an m-resolving set of <span class="math inline"><em>G</em></span>, if such a set exists. If <span class="math inline"><em>G</em></span> does not possess an m-resolving set, then we say that <span class="math inline"><em>G</em></span> has infinite multiset dimension. In this paper, we show that all cylindrical graphs <span class="math inline"><em>P</em><sub><em>m</em></sub> ▫ <em>C</em><sub><em>n</em></sub></span>, where <span class="math inline"><em>m</em>, <em>n</em> ≥ 3</span>, have finite multiset dimension. In particular, we show that md<span class="math inline">(<em>P</em><sub><em>m</em></sub> ▫ <em>C</em><sub><em>n</em></sub>) ≤ 4</span> if <span class="math inline"><em>m</em> ≥ 6</span> and <span class="math inline"><em>n</em> ≥ 3</span>, or if <span class="math inline"><em>m</em> ≥ 3</span> and <span class="math inline"><em>n</em> ≥ 12</span>. Moreover, if <span class="math inline"><em>m</em> ≥ 3</span> and <span class="math inline"><em>n</em> ≥ 8<em>m</em> + 1</span>, we show that <span class="math inline"><em>P</em><sub><em>m</em></sub> ▫ <em>C</em><sub><em>n</em></sub></span> has multiset dimension 3.
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| 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