Recognizing weighted and seeded disk graphs
Bibliographic record
Abstract
Disk intersection representations realize graphs by mapping vertices bijectively to disks in the plane such that two disks intersect each other if and only if the corresponding vertices are adjacent in the graph. If intersections are restricted to touching points of the boundaries, we call them disk contact representations. Deciding whether a vertex-weighted planar graph can be realized such that the disks' radii coincide with the vertex weights is known to be NP-hard for both contact and intersection representations. In this work, we reduce the gap between hardness and tractability by analyzing the problem for special graph classes. We show that in the contact scenario it remains NP-hard for outerplanar graphs with unit weights and for stars with arbitrary weights, strengthening the previous hardness results. On the positive side, we present constructive linear-time recognition algorithms for caterpillars with unit weights and for embedded stars with arbitrary weights. We also consider a version of the problem in which the disks of a representation are supposed to cover preassigned points, called seeds. We show that both for contact and intersection representations this problem is NP-hard for unit weights even if the given graph is a path. If the disks' radii are not prescribed, the problem remains NP-hard for trees in the contact scenario.
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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 | 0.000 | 0.003 |
| 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 | 0.001 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".