{"paper":{"title":"Three remarks on $\\mathbf{W_2}$ graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Carl Feghali, Malory Marin","submitted_at":"2023-07-28T14:12:22Z","abstract_excerpt":"Let $k \\geq 1$. A graph $G$ is $\\mathbf{W_k}$ if for any $k$ pairwise disjoint independent vertex subsets $A_1, \\dots, A_k$ in $G$, there exist $k$ pairwise disjoint maximum independent sets $S_1, \\dots, S_k$ in $G$ such that $A_i \\subseteq S_i$ for $i \\in [k]$. Recognizing $\\mathbf{W_1}$ graphs is co-NP-hard, as shown by Chv\\'atal and Slater (1993) and, independently, by Sankaranarayana and Stewart (1992). Extending this result and answering a recent question of Levit and Tankus, we show that recognizing $\\mathbf{W_k}$ graphs is co-NP-hard for $k \\geq 2$. On the positive side, we show that re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.15573","kind":"arxiv","version":2},"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/2307.15573/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"}