{"paper":{"title":"Clustered independence and bounded treewidth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Kolja Knauer, Torsten Ueckerdt","submitted_at":"2023-03-23T20:24:27Z","abstract_excerpt":"A set $S\\subseteq V$ of vertices of a graph $G$ is a $c$-clustered set if it induces a subgraph with components of order at most $c$ each, and $\\alpha_c(G)$ denotes the size of a largest $c$-clustered set. For any graph $G$ on $n$ vertices and treewidth $k$, we show that $\\alpha_c(G) \\geq \\frac{c}{c+k+1}n$, which improves a result of Dvo\\v{r}{\\'a}k and Wood [Innov.\\ Graph Theory, 2025], while we construct $n$-vertex graphs $G$ of treewidth $k$ with $\\alpha_c(G)\\leq \\frac{c}{c+k}n$. In the case $c\\leq 2$ or $k=1$ we prove the better lower bound $\\alpha_c(G) \\geq \\frac{c}{c+k}n$, which settles a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.13655","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2303.13655/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}