Bibliographic record
Abstract
We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph $G$ is a set $S$ of vertices such that each vertex is either in $S$ or has a neighbour in $S$. In a reconfiguration problem, the goal is to determine whether there exists a sequence of feasible solutions connecting given feasible solutions $s$ and $t$ such that each pair of consecutive solutions is adjacent according to a specified adjacency relation. Two dominating sets are adjacent if one can be formed from the other by the addition or deletion of a single vertex. For various values of $k$, we consider properties of $D_k(G)$, the graph consisting of a vertex for each dominating set of size at most $k$ and edges specified by the adjacency relation. Addressing an open question posed by Haas and Seyffarth, we demonstrate that $D_{Γ(G)+1}(G)$ is not necessarily connected, for $Γ(G)$ the maximum cardinality of a minimal dominating set in $G$. The result holds even when graphs are constrained to be planar, of bounded tree-width, or $b$-partite for $b \ge 3$. Moreover, we construct an infinite family of graphs such that $D_{γ(G)+1}(G)$ has exponential diameter, for $γ(G)$ the minimum size of a dominating set. On the positive side, we show that $D_{n-m}(G)$ is connected and of linear diameter for any graph $G$ on $n$ vertices having at least $m+1$ independent edges.
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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 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.008 | 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 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".