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 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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 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".