Efficient <i>(j, k)</i> -dominating functions
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Bibliographic record
Abstract
For positive integers j and k, an efficient (j, k)-dominating function of a graph G = (V, E) is a function f: V → {0, 1, 2,..., j} such that the sum of function values in the closed neighbourhood of every vertex equals k. The relationship between the existence of efficient (j, k)-dominating functions and various kinds of efficient dominating sets is explored. It is shown that if a strongly chordal graph has an efficient (j, k)-dominating function, then it has an efficient dominating set. Further, every efficient (j, k)-dominating function of a strongly chordal graph can be expressed as a sum of characteristic functions of efficient dominating sets. For j < k there are strongly chordal graphs with an efficient dominating set but no efficient (j, k)-dominating function. The problem of deciding whether a given graph has an efficient (j, k)-dominating function is shown to be NP-complete for all positive integers j and k, and solvable in polynomial time for strongly chordal graphs when j = k. By taking j = 1 we obtain NP-completeness of the problem of deciding whether a given graph has an efficient k-tuple dominating set for any fixed positive integer k. Finally, we consider efficient (2, 2)-dominating functions of trees. We describe a new constructive characterization of the trees with an efficient dominating set and a constructive characterization of the trees with two different efficient dominating sets. A number of open problems and questions are stated throughout the work.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it