Bibliographic record
Abstract
A hole in a graph is an induced subgraph which is a cycle of length at least four. A graph is chordal if it contains no holes. Following McKee and Scheinerman (1993), we define the chordality of a graph $G$ to be the minimum number of chordal graphs on $V(G)$ such that the intersection of their edge sets is equal to $E(G)$. In this paper we study classes of graphs of bounded chordality.In the 1970s, Buneman, Gavril, and Walter, proved independently that chordal graphs are exactly the intersection graphs of subtrees in trees. We generalize this result by proving that the graphs of chordality at most $k$ are exactly the intersection graphs of convex subgraphs of median graphs of tree-dimension$k$.A hereditary class of graphs $\mathcal{A}$ is $\chi$-bounded if there exists a function $ f\colon \mathbb{N}\rightarrow \mathbb{R}$ such that for every graph $G\in \mathcal{A}$, we have $\chi(G) \leq f(\omega(G))$. In 1960, Asplund and Grünbaum proved that the class of all graphs of boxicity at most two is $\chi$-bounded. In his seminal paper "Problems from the world surrounding perfect graphs," Gyárfás (1985), motivated by the above result, asked whether the class of all graphs of chordality at most two, which we denote by $\mathcal{C}\,{\mathop{\cap}\limits_{\raise.2ex\hbox{$\scriptstyle\bullet$}}}\,\mathcal{C}$, is $\chi$-bounded. We discuss a result of Felsner, Joret, Micek, Trotter and Wiechert (2017), concerning tree-decompositions of Burling graphs, which implies an answer to Gyárfás' question in the negative. We prove that two natural families of subclasses of $\mathcal{C}\,{\mathop{\cap}\limits_{\bullet}}\,\mathcal{C}$ are polynomially $\chi$-bounded.Finally, we prove that for every $k\geq 3$ the $k$-Chordality Problem, which asks to decide whether a graph has chordality at most $k$, is 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 | 0.001 | 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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".