{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:AV4HS7XGJYK64J464NZFVINITB","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0d8eee7c41fb397757d3973702feccbe76f787448b5360bd467d34247447339e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2018-06-04T13:51:29Z","title_canon_sha256":"d9d36749aac4b260ab3757f5549d3fbf0bde3672dce8232de7c23199c91b2bd3"},"schema_version":"1.0","source":{"id":"1806.01119","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.01119","created_at":"2026-05-18T00:14:17Z"},{"alias_kind":"arxiv_version","alias_value":"1806.01119v1","created_at":"2026-05-18T00:14:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.01119","created_at":"2026-05-18T00:14:17Z"},{"alias_kind":"pith_short_12","alias_value":"AV4HS7XGJYK6","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_16","alias_value":"AV4HS7XGJYK64J46","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_8","alias_value":"AV4HS7XG","created_at":"2026-05-18T12:32:13Z"}],"graph_snapshots":[{"event_id":"sha256:15ed0c8e412422c4a0a2430c49ce3ec34eb5d2a838a5bf19008a8d252c05dec7","target":"graph","created_at":"2026-05-18T00:14:17Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Finding cohesive subgraphs in a network is a well-known problem in graph theory. Several alternative formulations of cohesive subgraph have been proposed, a notable example being $s$-club, which is a subgraph where each vertex is at distance at most $s$ to the others. Here we consider the problem of covering a given graph with the minimum number of $s$-clubs. We study the computational and approximation complexity of this problem, when $s$ is equal to 2 or 3. First, we show that deciding if there exists a cover of a graph with three $2$-clubs is NP-complete, and that deciding if there exists a","authors_text":"Florian Sikora, Giancarlo Mauri, Italo Zoppis, Riccardo Dondi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2018-06-04T13:51:29Z","title":"Covering with Clubs: Complexity and Approximability"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.01119","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:15f68271ed29ad73dc19d703405cb65030ca08407753e62e6c3bd93cd78fee1d","target":"record","created_at":"2026-05-18T00:14:17Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0d8eee7c41fb397757d3973702feccbe76f787448b5360bd467d34247447339e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2018-06-04T13:51:29Z","title_canon_sha256":"d9d36749aac4b260ab3757f5549d3fbf0bde3672dce8232de7c23199c91b2bd3"},"schema_version":"1.0","source":{"id":"1806.01119","kind":"arxiv","version":1}},"canonical_sha256":"0578797ee64e15ee279ee3725aa1a89862eeebd7bc8bd329d6a7851708f88341","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0578797ee64e15ee279ee3725aa1a89862eeebd7bc8bd329d6a7851708f88341","first_computed_at":"2026-05-18T00:14:17.413122Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:17.413122Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OFmne1hycIro0lDvA5+70qvgVmewSIRmKzMjXFSoCesSjCQy2bJjVEsx9lO7fiOrIPw9OBgvfTlonSzSBYkuDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:17.413862Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.01119","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:15f68271ed29ad73dc19d703405cb65030ca08407753e62e6c3bd93cd78fee1d","sha256:15ed0c8e412422c4a0a2430c49ce3ec34eb5d2a838a5bf19008a8d252c05dec7"],"state_sha256":"e61ce2504c79becfecce5a1528ba66e811714d2c7ab0fe4d526a6f3102ccdf53"}