Bibliographic record
Abstract
Let F be a set of ordered patterns, i.e., graphs whose vertices are linearly ordered. An F-free ordering of the vertices of a graph H is a linear ordering of V(H) such that none of patterns in F occurs as an induced ordered subgraph. We denote by ORD(F) the decision problem asking whether an input graph admits an F-free ordering; we also use ORD(F) to denote the class of graphs that do admit an F-free ordering. It was observed by Damaschke (and others) that many natural graph classes can be described as ORD(F) for sets F of small patterns (with three or four vertices). Damaschke also noted that for many sets F consisting of patterns with three vertices, ORD(F) is polynomial-time solvable by known algorithms or their simple modifications. We complete the picture by proving that all these problems can be solved in polynomial time. In fact, we provide a single master algorithm, i.e., we solve in polynomial time the problem $ORD_3$ in which the input is a set F of patterns with at most three vertices and a graph H, and the problem is to decide whether or not H admits an F-free ordering of the vertices. Our algorithm certifies non-membership by a forbidden substructure, and thus provides a single forbidden structure characterization for all the graph classes described by some ORD(F) with F consisting of patterns with at most three vertices. Many of the problems ORD(F) with F consisting of larger patterns have been shown to be NP-complete by Duffus, Ginn, and Rodl, and we add two simple examples. We also discuss a bipartite version of the problem, BORD(F), in which the input is a bipartite graph H with a fixed bipartition of the vertices, and we are given a set F of bipartite patterns. We also describe some examples of digraph ordering problems and algorithms. We conjecture that for every set F of forbidden patterns, ORD(F) is either polynomial or NP-complete.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.004 | 0.005 |
| 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".