{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:BJ6UTTYZXCHLUY7H4UVWFLKC3N","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":"b01c672b7bf9fbdfc324ba45f64022df14cc68cd7ad98c9ea576591088d8cd02","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-02-28T07:24:17Z","title_canon_sha256":"f29c38befb305b0d0f0663f14452590394cc43dd8ab402a7a515060ce4a71b9f"},"schema_version":"1.0","source":{"id":"1503.00091","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.00091","created_at":"2026-05-18T02:25:55Z"},{"alias_kind":"arxiv_version","alias_value":"1503.00091v1","created_at":"2026-05-18T02:25:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.00091","created_at":"2026-05-18T02:25:55Z"},{"alias_kind":"pith_short_12","alias_value":"BJ6UTTYZXCHL","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"BJ6UTTYZXCHLUY7H","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"BJ6UTTYZ","created_at":"2026-05-18T12:29:14Z"}],"graph_snapshots":[{"event_id":"sha256:10a6adfc5aab8d14766c8b0113d65a256e8b8cbe690d8dcd40de274b3bda56d6","target":"graph","created_at":"2026-05-18T02:25:55Z","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":"Let $G$ be a finite undirected graph. A vertex {\\em dominates} itself and all its neighbors in $G$. A vertex set $D$ is an {\\em efficient dominating set} (\\emph{e.d.}\\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \\emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.\\ in $G$, is known to be \\NP-complete even for very restricted graph classes such as $P_7$-free chordal graphs. The ED problem on a graph $G$ can be reduced to the Maximum Weight Independent Set (MWIS) problem on the square of $G$. The complexity of the ED proble","authors_text":"Andreas Brandst\\\"adt, Elaine M. Eschen, Erik Friese","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-02-28T07:24:17Z","title":"Efficient Domination for Some Subclasses of $P_6$-Free Graphs in Polynomial Time"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.00091","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":"sha256:82814508b13963bfc76b3d7d8c49c1ef6582d0454affc621697756a9e6787add","target":"record","created_at":"2026-05-18T02:25:55Z","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":"b01c672b7bf9fbdfc324ba45f64022df14cc68cd7ad98c9ea576591088d8cd02","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-02-28T07:24:17Z","title_canon_sha256":"f29c38befb305b0d0f0663f14452590394cc43dd8ab402a7a515060ce4a71b9f"},"schema_version":"1.0","source":{"id":"1503.00091","kind":"arxiv","version":1}},"canonical_sha256":"0a7d49cf19b88eba63e7e52b62ad42db4f420fbbbeafd89d01d07f66199c7648","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0a7d49cf19b88eba63e7e52b62ad42db4f420fbbbeafd89d01d07f66199c7648","first_computed_at":"2026-05-18T02:25:55.165641Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:25:55.165641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yT8Jvk7AfjA7N0QxHwsJ+TojJ6euOmmtINUN8pPd6l8WLfTe1wDufvmfsEMy//HL3qbDfQcffc0jE7KgXlenBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:25:55.166081Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.00091","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:82814508b13963bfc76b3d7d8c49c1ef6582d0454affc621697756a9e6787add","sha256:10a6adfc5aab8d14766c8b0113d65a256e8b8cbe690d8dcd40de274b3bda56d6"],"state_sha256":"b7f5ac8230af041b25cfaf2e6150e82b24299f45f4025466c6f80529fbddf66c"}