<i>K</i><sub>4</sub>‐free and ‐free Planar Matching Covered Graphs
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Bibliographic record
Abstract
Abstract A bi‐subdivision of a graph J is a graph H obtained from J by subdividing each of its edges by inserting an even number of vertices. A matching covered subgraph H of a matching covered graph G is conformal if has a perfect matching. Using the theory of ear decompositions, Lovász (Combinatorica, 3 (1983), 105–117) showed that every nonbipartite matching covered graph has a conformal subgraph which is either a bi‐subdivision of K 4 or of . (The graph is the triangular prism.) A matching covered graph is K 4 ‐ based if it contains a bi‐subdivision of K 4 as a conformal subgraph; otherwise it is K 4 ‐ free . ‐ based and ‐ free graphs are analogously defined. The result of Lovász quoted above implies that any nonbipartite matching covered graph is either K 4 ‐based or ‐based (or both). The problem of deciding which matching covered graphs are K 4 ‐based and which are ‐based is, in general, unsolved. In this paper, we present a solution to this classification problem in the special case of planar graphs. In Section 2, we show that a matching covered graph is K 4 ‐free ( ‐free) if and only if each of its bricks is K 4 ‐free ( ‐free). In Section 5, we show that a planar brick is K 4 ‐free if and only if it has precisely two odd faces. In Section 6, we determine the list of all ‐free planar bricks; apart from one exception, it consists of two infinite families of bricks. The principal tool we use for proving our results is the brick generation procedure established by Norine and Thomas (J Combin Theory Ser B, 97 (2007), 769–817).
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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 | 0.005 | 0.001 |
| 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 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.003 | 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