Recognizing <i>k</i> -Leaf Powers in Polynomial Time, for Constant <i>k</i>
Why this work is in the frame
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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 (and u ≠ v ). 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 article, we present an algorithm that decides whether a graph G is a k -leaf power in time O ( n f(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 long-standing open problem, it is our hope that the structural results presented in this work can lead to further results on k -leaf powers and related classes.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it