Distance between two k-sets and Path-Systems Extendibility.
Bibliographic record
Abstract
Given a simple graph G on n vertices, let σ2(G) be the minimum sum of the degrees of any two non adjacent vertices. The graph G is said to be connected if any two distinct vertices may be joined by a path. It is easy to see that if σ2(G) ≥ n − 1 then G is not only connected, but we can choose the connecting path to be of size at most two. Ore [4] proved that if σ2(G) ≥ n+ 1 we may always choose this path to cover all the vertices of G. In this paper we extend these results to systems of vertex disjoint paths connecting two vertex k-sets of G. 1 Preliminaries In this paper, G = (V,E) will denote a simple loopless graph with |G| = |V (G)| = n. The order and the size of a graph are respectfully the number of vertices and the number of edges in this graph. Definitions and notation that are not found here may be found in [2]. Let u, v ∈ V (G). If u 6= v, a [u, v]-path is a subgraph P of G constituted of a sequence of distinct vertices u = z1, z2, . . . , zp−1, zp = v along with edges between zi and zi+1 (for all 1 ≤ i ≤ p−1). We will consider a vertex u to be a [u, u]-path of order one; in this case we say the path is singular. The size of a path is the number of it’s edges.
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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 | 0.001 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".