Direct Planar Tree Transformation and Counterexample
Bibliographic record
Abstract
We consider the problem of planar spanning tree transformation in a two-dimensional plane. Given two planar trees T1 and T2 drawn on a set S of n points in general position in the plane, the problem is to transform T1 into T2 by a sequence of simple changes called edge-flips or just flips. A flip is an operation by which one edge e of a geometric object is removed and an edge f (f = e) is inserted such that the resulting object belongs to the same class as the original object. Generally, for geometric transformation, the usual technique is to rely on some ‘canonical ’ object which can be obtained by making simple changes to the initial object and then doing the reverse operations that transform the canonical object to the desired object. In this paper, we present a technique for such transformation that does not rely on any canonical tree. It is shown that T1 and T2 can be transformed into each other by at most n − 1 + k flips (k ≥ 0) when S is in convex position and we also show results when S is in general position. We provide cases where the approach performs an optimal number of flips. A counterexample is given to show that if |T1 \\ T2 | = k then they cannot be transformed to one another by at most k flips. 1
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 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".