Bibliographic record
Abstract
If G is a bipartite graph with bipartition (X, Y ), a subset S of X is called a one-sided dominating set if every vertex y ∈ Y is adjacent to some x ∈ S. If S is minimal as a one-sided dominating set (i.e. if it has no proper subset which is also a one-sided dominating set, ) it is called a bipartite dominating set (see [4],[5], and [6]). We study bipartite dominating sets in hypercubes. Definition 1 Let G be a bipartite graph with bipartition (X,Y ). A subset S of X is called a one-sided dominating set if every vertex y ∈ Y is adjacent to some x ∈ S, i.e. if N(S) = Y . S is a minimal onesided dominating set if no proper subset of S is a one-sided dominating set. It is a minimum one-sided dominating set if no one-sided dominating set contained in X has smaller cardinality. In that case, S is called a bipartite dominating set. Bipartite dominating sets have been studied by Haynes, Hedetniemi, and Slater [4] and by Hedetniemi and Laskar [5], [6]. Remark 1 A subset S of X is a one-sided dominating set ⇔ the only maximal independent set containing S is X. Notation. For any graph G, γ(G) denotes the minimum size of a dominating set in G. We denote by Qn the n-dimensional hypercube. Its bipartition (X,Y ) is given by X = {x ∈ Qn | wt(x) is even}, Y = {y ∈ Qn | wt(y) is odd} where wt(z), the weight of z, is the number of 1’s in the n-tuple z. Alternatively, if we think of the vertices of Qn as the subsets of {1, 2, . . . , n}, X consists of the subsets of even cardinality, and Y consists of the subsets of odd cardinality. We will also at times consider Qn to be a group under component-wise addition of n-tuples (or, if the vertices are thought of as subsets of {1, 2, . . . , n}, then under the operation of symmetric difference). The next proposition basically restates the Hamming Bound (see [9], p. 413), for Qn for single-error-correcting codes. Department of Mathematics, Northeastern University, Boston, MA 02115 (ramras@neu.edu), Tel: 617-373-5651, Fax: 617-373-5658.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.004 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".