Constrained Outer-String Representations
Bibliographic record
Abstract
An outer-string representation of a graph is an intersection representation in which each vertex is represented by a curve that is contained in the unit disk and has at least one endpoint on the boundary of the unit disk. In an outer-1-string representation the curves representing any two vertices are in addition allowed to intersect at most once. In this paper, we consider the following constrained version: Given a graph G plus a cyclic order v_1,…,v_n of the vertices in G, test whether G has an outer-string or an outer-1-string representation in which the curves representing v_1,…,v_n intersect the boundary of the unit disk in this order. We first show that a graph has an outer-string representation for all possible cyclic orders of the vertices if and only if the graph is the complement of a chordal graph. Then we turn towards the situation where one particular cyclic order of the vertices is fixed. We characterize the chordal graphs admitting a constrained outer-string representation and the trees and cycles admitting a constrained outer-1-string representation. The characterizations yield polynomial-time recognition and construction algorithms; in the case of outer-1-string representations the run time is linear. We also show how to decide in polynomial time whether an arbitrary graph admits a constrained L-shaped outer-1-string representation. In an L-shaped representation the curves are 1-bend orthogonal polylines anchored on a horizontal line, and they are contained in the half-plane below that line. However, not even all paths with a constrained outer-1-string representation admit one with L-shapes. We show that 2-bend orthogonal polylines are sufficient for trees and cycles with a constrained outer-1-string representation.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.015 | 0.003 |
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".