Some Structural Properties of a Least Central Subtree of a Tree
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Bibliographic record
Abstract
We consider the graph center problem in the joinsemilattice L(T ) of all subtrees of a tree T . A subtree S of a tree T is a central subtree of T if S has the minimum eccentricity in the joinsemilattice. The graph center of the joinsemilattice is the set of all central subtrees. A central subtree with the minimum number of points is a least central subtree of a tree T . Thus least central subtrees of T are, in some sense, the best possible connected substructures of T among all connected substructures. We show that every tree is a unique least central subtree of some larger tree. Our main result points out the importance of the cardinality of the nodes of degree two. Low cardinality guarantees uniqueness and explicit construction for the least central subtree.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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