On the Tractability of Covering a Graph with 2-Clubs
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Bibliographic record
Abstract
Abstract Covering a graph with cohesive subgraphs is a classical problem in theoretical computer science, for example when the cohesive subgraph model considered is a clique. In this paper, we consider as a model of cohesive subgraph the 2-clubs, which are induced subgraphs of diameter at most 2. We prove new complexity results on the $$\mathsf {Min~2\text {-}Club~Cover}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Min</mml:mi><mml:mspace/><mml:mn>2</mml:mn><mml:mtext>-</mml:mtext><mml:mi>Club</mml:mi><mml:mspace/><mml:mi>Cover</mml:mi></mml:mrow></mml:math> problem, a variant recently introduced in the literature which asks to cover the vertices of a graph with a minimum number of 2-clubs. First, we answer an open question on the decision version of $$\mathsf {Min~2\text {-}Club~Cover}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Min</mml:mi><mml:mspace/><mml:mn>2</mml:mn><mml:mtext>-</mml:mtext><mml:mi>Club</mml:mi><mml:mspace/><mml:mi>Cover</mml:mi></mml:mrow></mml:math> that asks if it is possible to cover a graph with at most two 2-clubs, and we prove that it is W[1]-hard when parameterized by the distance to a 2-club. Then, we consider the complexity of $$\mathsf {Min~2\text {-}Club~Cover}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Min</mml:mi><mml:mspace/><mml:mn>2</mml:mn><mml:mtext>-</mml:mtext><mml:mi>Club</mml:mi><mml:mspace/><mml:mi>Cover</mml:mi></mml:mrow></mml:math> on some graph classes. We prove that $$\mathsf {Min~2\text {-}Club~Cover}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Min</mml:mi><mml:mspace/><mml:mn>2</mml:mn><mml:mtext>-</mml:mtext><mml:mi>Club</mml:mi><mml:mspace/><mml:mi>Cover</mml:mi></mml:mrow></mml:math> remains NP-hard on subcubic planar graphs, W[2]-hard on bipartite graphs when parameterized by the number of 2-clubs in a solution, and fixed-parameter tractable on graphs having bounded treewidth.
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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 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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 it