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 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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".