Recognizing a totally odd K4-subdivision, parity 2-disjoint rooted paths and a parity cycle through specified elements
Bibliographic record
Abstract
A totally odd K4-subdivision is a subdivision of K4 where each subdivided edge has odd length. The recognition of a totally odd K4-subdivision plays an important role in both graph theory and combinatorial optimization. Sewell and Trotter [53], Zang [63] and Thomassen [60] independently conjectured the existence of a polynomial time recognition algorithm. In this paper, we give the first polynomial time algorithm for solving this problem.We also study the the parity two disjoint rooted paths problem where we determine if there exists two vertex disjoint paths of a specified parity between two pairs of terminals.Using a similar technique, we give an O(|E(G)||V(G)|α(|E(G)|,|V(G)|)) algorithm for the parity two disjoint rooted paths problem on an input graph G, where α(|E(G)|,|V(G)|) is the inverse of the Ackermann function. We note that this clearly gives an algorithm for the well-known non-parity version of the two disjoint rooted paths problem [19, 50, 52, 55, 58].We then extend our approach to give a polynomial time algorithm which determines, for any fixed k, whether there exists a cycle of a given parity through k independent input edges.This generalizes the non-parity version of the algorithm in [22]. Thomassen [61] gave a polynomial algorithm for the case k = 2 and hoped to use this algorithm to recognize a totally odd K4-subdivision. Our algorithm runs in O(|E(G)||V(G)|α(|E(G)|,|V(G)|)) for any fixed k.Finally, we give an O(|V(G)|2 + |E(G)|α(|E(G)|,|V(G)|log|V(G)|)) algorithm to decide whether a graph contains k disjoint paths from A to B (with |A| = |B| = k) that are not all of the same parity.This answers a conjecture of Thomassen [60]. This problem arises from the study of totally odd-K4-subdivisions in 3-connected graphs [60].
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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| 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".