k-Leaf Powers Cannot Be Characterized by a Finite Set of Forbidden Induced Subgraphs for k ≥ 5
Bibliographic record
Abstract
A graph G = (V,E) is a k-leaf power if there is a tree T whose leaves are the vertices of G, with the property that a pair of distinct leaves u and v share an edge in G if and only if they are distance at most k apart in T. For k ≤ 4, it is known that there exists a finite set F_k of graphs such that the class ℒ(k) of k-leaf power graphs is characterized as the set of strongly chordal graphs that do not contain any graph in F_k as an induced subgraph. We prove no such characterization holds for k ≥ 5. That is, for any k ≥ 5, there is no finite set F_k of graphs such that ℒ(k) is equivalent to the set of strongly chordal graphs that do not contain as an induced subgraph any graph in F_k.
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How this classification was reachedexpand
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) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| 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 | 0.000 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".