Contraction Obstructions for Connected Graph Searching
Bibliographic record
Abstract
We consider the connected variant of the classic mixed search game where, in each search step, cleaned edges form a connected subgraph. We consider graph classes with bounded connected (and monotone) mixed search number and we deal with the question whether the obstruction set, with respect of the contraction partial ordering, for those classes is finite. In general, there is no guarantee that those sets are finite, as graphs are not well quasi ordered under the contraction partial ordering relation. In this paper we provide the obstruction set for $k=2$, where $k$ is the number of searchers we are allowed to use. This set is finite, it consists of 177 graphs and completely characterises the graphs with connected (and monotone) mixed search number at most 2. Our proof reveals that the "sense of direction" of an optimal search searching is important for connected search which is in contrast to the unconnected original case. We also give a double exponential lower bound on the size of the obstruction set for the classes where this set is finite.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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".