A Clique-Based Separator for Intersection Graphs of Geodesic Disks in $$\mathbb {R}^2$$
Bibliographic record
Abstract
Abstract Let d be a (well-behaved) shortest-path metric defined on a path-connected subset of $$\mathbb {R}^2$$ and let $$\mathcal {D}=\{D_1,\ldots,D_n\}$$ be a set of geodesic disks with respect to the metric d . We prove that $$\mathcal {G}^{\times }(\mathcal {D})$$ , the intersection graph of the disks in $$\mathcal {D}$$ , has a clique-based separator consisting of $$O(n^{3/4+\varepsilon })$$ cliques. This significantly extends the class of objects whose intersection graphs have small clique-based separators. Our clique-based separator yields an algorithm for q - Coloring that runs in time $$2^{O(n^{3/4+\varepsilon })}$$ , assuming the boundaries of the disks $$D_i$$ can be computed in polynomial time. We also use our clique-based separator to obtain a simple, efficient, and almost exact distance oracle for intersection graphs of geodesic disks. Our distance oracle uses $$O(n^{7/4+\varepsilon })$$ storage and can report the hop distance between any two nodes in $$\mathcal {G}^{\times }(\mathcal {D})$$ in $$O(n^{3/4+\varepsilon })$$ time, up to an additive error of one. So far, distance oracles with an additive error of one that use subquadratic storage and sublinear query time were not known for such general graph classes.
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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 | 0.001 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.007 | 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".