{"paper":{"title":"Graphs without two vertex-disjoint $S$-cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Minjeong Kang, Myounghwan Lee, O-joung Kwon","submitted_at":"2019-08-24T01:22:50Z","abstract_excerpt":"Lov\\'asz (1965) characterized graphs without two vertex-disjoint cycles, which implies that such graphs have at most three vertices hitting all cycles. In this paper, we ask whether such a small hitting set exists for $S$-cycles, when a graph has no two vertex-disjoint $S$-cycles. For a graph $G$ and a vertex set $S$ of $G$, an $S$-cycle is a cycle containing a vertex of $S$.\n  We provide an example $G$ on $21$ vertices where $G$ has no two vertex-disjoint $S$-cycles, but three vertices are not sufficient to hit all $S$-cycles. On the other hand, we show that four vertices are enough to hit al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.09065","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/1908.09065/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"}