The Forcing Strong Metric Dimension of a Graph
Why this work is in the frame
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Bibliographic record
Abstract
For any two vertices u, v in a connected graph G, the interval I(u, v) consists of all vertices which are lying in some u − v shortest path in G. A vertex x in a graph G strongly resolves a pair of vertices u, v if either u ∈ I(x, v) or v ∈ I(x, u). A set of vertices W of V (G) is called a strong resolving set if every pair of vertices of G is strongly resolved by some vertex of W. The minimum cardinality of a strong resolving set in G is called the strong metric dimension number of G and it is denoted by sdim(G). For a strong resolving set W of G, a subset S of W is called the forcing subset of W if W is the unique strong resolving set containing S. The forcing number f(W, sdim(G)) of W in G is the minimum cardinality of a forcing subset for W, while the forcing strong metric dimension, fsdim(G), of G is the smallest forcing number among all strong resolving sets of G. The forcing strong metric dimensions of some well-known graphs are determined. It is shown that for any positive integers a and b, with 0 ≤ a ≤ b, there is nontrivial connected graph G with sdim(G) = b and fsdim(G) = a if and only if {a, b} not equal to {0, 1}.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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