Recognizing <i>k</i>-leaf powers in polynomial time, for constant <i>k</i>
Bibliographic record
Abstract
A graph G is a k-leaf power if there exists a tree T whose leaf set is V (G), and such that uv ∊ E(G) if and only if the distance between u and v in T is at most k. The graph classes of k-leaf powers have several applications in computational biology, but recognizing them has remained a challenging algorithmic problem for the past two decades. The best known result is that 6-leaf powers can be recognized in polynomial time. In this paper, we present an algorithm that decides whether a graph G is a k-leaf power in time O(nf(k)) for some function f that depends only on k (but has the growth rate of a power tower function). Our techniques are based on the fact that either a k-leaf power has a corresponding tree of low maximum degree, in which case finding it is easy, or every corresponding tree has large maximum degree. In the latter case, large degree vertices in the tree imply that G has redundant substructures which can be pruned from the graph. In addition to solving a longstanding open problem, we hope that the structural results presented in this work can lead to further results on k-leaf powers.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.015 | 0.007 |
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".