Bibliographic record
Abstract
The domatic number of a graph is the maximum number of pairwise disjoint dominating sets admitted by the graph. We introduce a game based around this graph invariant. The domatic number game is played on a graph \(G\) by two players, Alice and Bob, who take turns selecting a vertex and placing it into one of \(k\) sets. Alice is trying to make each of these sets into a dominating set of \(G\) while Bob’s goal is to prevent this from being accomplished. The maximum \(k\) for which Alice can achieve her goal when both players are playing optimal strategies, is called the game domatic number of \(G\). There are two versions of the game and two resulting invariants depending on whether Alice or Bob is the first to play. We prove several upper bounds on these game domatic numbers of arbitrary graphs and find the exact values for several classes of graphs including trees, complete bipartite graphs, cycles and some narrow grid graphs. We pose several open problems concerning the effect of standard graph operations on the game domatic number as well as a vexing question related to the monotonicity of the number of sets available to Alice.
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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 | 0.002 | 0.009 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.009 | 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".