Maximum boundary independent broadcasts in graphs and trees
Bibliographic record
Abstract
A broadcast on a connected graph G is a function f : V ( G )→{0, 1, ..., d i a m ( G )} such that f ( v )≤ e ( v ) (the eccentricity of v ) for all v ∈ V . If d G ( u , v )≥ f ( u )+ f ( v ) for any pair of vertices u , v with f ( u )>0 and f ( v )>0 , the broadcast is said to be boundary independent. We show that the maximum weight α b n ( G ) of a boundary independent broadcast can be bounded in terms of the independence number α ( G ) , and prove that the maximum boundary independent broadcast problem is NP-hard. We investigate bounds on α b n ( T ) when T is a tree in terms of its order and the number of vertices of degree at least 3, and determine a sharp upper bound on α b n ( T ) when T is a caterpillar, giving α b n ( T ) exactly for certain families of caterpillars. We conclude by describing a polynomial-time algorithm to determine α b n ( T ) for a given tree T .
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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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".