<i>K</i><sub>4</sub>‐free and ‐free Planar Matching Covered Graphs
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 K4 or of . (The graph is the triangular prism.) A matching covered graph is K4‐based if it contains a bi‐subdivision of K4 as a conformal subgraph; otherwise it is K4‐free. ‐based and ‐free graphs are analogously defined. The result of Lovász quoted above implies that any nonbipartite matching covered graph is either K4‐based or ‐based (or both). The problem of deciding which matching covered graphs are K4‐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 K4‐free ( ‐free) if and only if each of its bricks is K4‐free ( ‐free). In Section 5, we show that a planar brick is K4‐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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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.008 | 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".