Graph homomorphism reconfiguration and frozen H‐colorings
Bibliographic record
Abstract
Abstract For a fixed graph H, the reconfiguration problem for H‐colorings (ie, homomorphisms to H) asks: given a graph G and two H‐colorings and of G, does there exist a sequence of H‐colorings such that , , and for every and ? If the graph G is loop‐free, then this is the equivalent to asking whether it possible to transform into by changing the color of one vertex at a time such that all intermediate mappings are H‐colorings. In the affirmative, we say that reconfigures to . Currently, the complexity of deciding whether an H‐coloring reconfigures to an H‐coloring is only known when H is a clique, a circular clique, a ‐free graph, or in a few other cases which are easily derived from these. We show that this problem is PSPACE‐complete when H is an odd wheel. An important notion in the study of reconfiguration problems for H‐colorings is that of a frozen H‐coloring; that is, an H‐coloring such that does not reconfigure to any H‐coloring such that . We obtain an explicit dichotomy theorem for the problem of deciding whether a given graph G admits a frozen H‐coloring. The hardness proof involves a reduction from a constraint satisfaction problem which is shown to be nondeterministic polynomial time NP‐complete by establishing the nonexistence of a certain type of polymorphism.
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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 | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".