Bibliographic record
Abstract
The bicontraction of a vertex $v$ of degree two in a graph, with precisely two neighbours $v_1$ and $v_2$, consists of shrinking the set $\{v_1,v,v_2\}$ to a single vertex. The retract of a matching covered graph $G$, denoted by $\widehat{G}$, is the graph obtained from $G$ by repeatedly bicontracting vertices of degree two. Up to isomorphism, the retract of a matching covered graph $G$ is unique. If $G$ is a brace on six or more vertices, an edge $e$ of $G$ is thin if $\widehat{G-e}$ is a brace. A thin edge $e$ in a simple brace $G$ is strictly thin if $\widehat{G-e}$ is a simple brace. Theorems concerning the existence of strictly thin edges have been used (implicitly by McCuaig (Pólya's Permanent Problem, Electron. J. of Combin., 11, 2004) and explicitly by the authors (On the Number of Perfect Matchings in a Bipartite Graph, SIAM J. Discrete Math., 27, 940-958, 2013)) as inductive tools for establishing properties of braces.Let $G$ and $J$ be two distinct braces, where $G$ is of order six or more and $J$ is a simple matching minor of $G$. It follows from a theorem of McCuaig (Brace Generation, J. Graph Theory, 38, 124-169, 2001) that $G$ has a thin edge $e$ such that $J$ is a matching minor of $G-e$. In Section 2, we give an alternative, and simpler proof, of this assertion. Our method of proof lends itself to proving stronger results concerning thin edges.Let ${\cal G}^+$ denote the family of braces consisting of all prisms, all Möbius ladders, all biwheels, and all extended biwheels. Strengthening another result of McCuaig on brace generation, we show that every simple brace of order six or more which is not a member of ${\cal G}^+$ has at least two strictly thin edges. We also give examples to show that this result is best possible.
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How this classification was reachedexpand
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 | 0.003 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".