Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Graph H is a root of graph G if there exists a natural number k such that xy ∈ E(G) ↔ dH(x, y) ≤ k where dH(x, y) is the length of a shortest path in H from x to y. In such a case, H is a k-th root of G and we write G = Hk and call G the k-th power of H. Motwani and Sudan proved that it is NP-complete to recognize squares of graphs and believed it is also NP-complete to recognize squares of bipartite graphs. In this paper, we show, rather surprisingly, that squares of bipartite graphs can be recognized in polynomial time. Also, we show that counting the number of different bipartite square roots of a graph can be done in polynomial time although this number could be exponential in the size of the input graph. Furthermore, we can generate all bipartite roots of a graph G in time O(max{Δ(G) · M, r(G)}) where Δ(G) is the maximum degree of G, M is the time complexity to do matrix multiplication, and r(G) is the number of different bipartite square roots of G. By using the tools developed, we are able to give a new and simpler linear time algorithm to recognize squares of trees and a new algorithmic proof that tree square roots, when they exist, are unique up to isomorphism. Finally, we prove the NP-completeness of recognition of cubes of bipartite graphs.
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.
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 | 0.001 | 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 it