Bounds on the metric and partition dimensions of a graph
Bibliographic record
Abstract
Given a graph G, we say S µ V (G) is resolving if for each pair of distinct u;v 2 V (G) there is a vertex x in S where d(u;x) 6 d(v;x). The metric dimension of G is the minimum cardinality of all resolving sets. For w 2 V (G), the distance from w to S, denoted d(w;S), is the minimum distance between w and the vertices of S. Given P = fP1;P2;:::;Pkg an ordered partition of V (G) we say P is resolving if for each pair of distinct u;v 2 V (G) there is a part Pi where d(u;Pi) 6 d(v;Pi). The partition dimension is the minimum order of all resolving partitions. In this paper we consider relationships between metric dimension, partition dimension, diameter, and other graph parameters. We construct \universal examples of graphs with given partition dimension, and we use these to provide bounds on various graph parameters based on metric and partition dimensions. We form † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † ¢¢¢ ¢¢¢ ¢¢¢ † † † † † ¢¢¢ | }
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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 | 0.003 | 0.029 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.004 | 0.003 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.004 | 0.012 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.003 | 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".