Bibliographic record
Abstract
For a connected graph G of order n, a set S of vertices of G is a geodetic set of G if each vertex n of G lies on a x-y geodesic for some elements x and y in S. The minimum cardinality of a geodetic set of G is defined as the geodetic number of G, denoted by g(G). A geodetic set of cardinality g(G) is called a g-set of G. A set S of vertices of a connected graph G is an open geodetic set of G if for each vertex n in G, either n is an extreme vertex of G and n S; or n is an internal vertex of an x-y geodesic for some x,yS. An open geodetic set of minimum cardinality is a minimum open geodetic set and this cardinality is the open geodetic number, og(G). An open geodetic set S in a connected graph G is called a minimal open geodetic set if no proper subset of S is an open geodetic set of G. The upper open geodetic number og+(G) of G is the maximum cardinality of a minimal open geodetic set of G. It is shown that, for a connected graph G of order n, og(G)=n, if and only if og+(G)=n, and also that og(G)=3 if any only if og+(G)=3. It is shown that for positive integers a and b with 4 ≤ a ≤ b, there exists a connected graph G with og(G) =a and og+(G)=b. Also, it is shown that for positive integers a, b, c with 4 ≤ a ≤ b ≤ c and b ≤ 3a, there exists a connected graph G with g(G)=a, og(G)=b and og+(G)= c.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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".