Four terminal planar Delta-Wye reducibility via rooted K2,4 minors
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Bibliographic record
Abstract
A graph with four special vertices (called terminals) is wye-delta reducible if we can obtain a graph on four vertices by a sequence of wye-delta and delta-wye operations and series-parallel reductions, none of which is allowed to remove any of the terminals. A good characterization of wye-delta reducible 3-connected planar graphs with four terminals is given. The proofs yield an O(n2) time algorithm that either exhibits an obstruction to the 4-terminal reducibility or returns a sequence of wye-delta operations and series-parallel reductions that reduce the input graph to a subgraph of K4 whose vertices are the terminals.We also discuss terminal wye-delta reducibility when a mixture of vertices and faces are treated as terminals. It is also shown that a sufficiently connected cubic graph is wye-delta reducible if and only if it does not contain the Petersen graph as a minor.The main ingredient in the proofs is a good characterization of planar graphs with four terminals that do not admit a rooted K2,4 minor with the four terminals corresponding to the roots on the large side of the bipartition of K2,4. Up to small connectivity reductions, cases without the rooted minor fall into five structural cases that lead to a polynomial-time algorithm for recognition of these graphs and construction of rooted K2,4 minors. This result is of independent interest in structural graph theory.
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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 | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.000 |
| 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