{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:D3OBIFGIAMDWS44NDAC5MYGWBN","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":"322a314f5c9a75aa24bdece6cbc4112fa5490d110f7e6179a2a3ebba32d8ec49","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-07T09:11:24Z","title_canon_sha256":"c58f36ff911e85a57721d2c97c49ce71c1d2e28f3cb6b3aed3dfc9224b17c942"},"schema_version":"1.0","source":{"id":"1804.02532","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.02532","created_at":"2026-05-18T00:18:59Z"},{"alias_kind":"arxiv_version","alias_value":"1804.02532v1","created_at":"2026-05-18T00:18:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.02532","created_at":"2026-05-18T00:18:59Z"},{"alias_kind":"pith_short_12","alias_value":"D3OBIFGIAMDW","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"D3OBIFGIAMDWS44N","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"D3OBIFGI","created_at":"2026-05-18T12:32:19Z"}],"graph_snapshots":[{"event_id":"sha256:30dcbd43b4076ff0f4c61308084d36867c16df08076f4140ab1c6fe514ebccb4","target":"graph","created_at":"2026-05-18T00:18:59Z","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":"A subset $D$ of vertices of a graph $G$ is a \\textit{dominating set} if for each $u\\in V(G)\\setminus D$, $u$ is adjacent to some vertex $v\\in D$. The \\textit{dominating number}, $\\gamma(G)$ of $G$, is the minimum cardinality of a dominating set of $G$. A set $D\\subseteq V(G)$ is a \\textit{total dominating set} if for each $u\\in V(G)$, $u$ is adjacent to some vertex $v\\in D$. the The \\textit{total dominating number}, $\\gamma_t(G)$ of $G$, is the minimum cardinality of a total dominating set of $G$. For an even integer $n\\ge2$ and $1\\le\\Delta\\le\\lfloor\\log_2n\\rfloor$, a \\textit{Kn\\\"odel graph} $","authors_text":"Doost Ali Mojdeh, Esmaeil Nazari, Nader Jafari Rad, Seyed Reza Musawi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-07T09:11:24Z","title":"Total domination in cubic Kn\\\"odel graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.02532","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":"sha256:24024df5456ac6085db9562d87afb41cf8e07137bbbf7e0415d6f5d670e5b8ae","target":"record","created_at":"2026-05-18T00:18:59Z","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":"322a314f5c9a75aa24bdece6cbc4112fa5490d110f7e6179a2a3ebba32d8ec49","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-07T09:11:24Z","title_canon_sha256":"c58f36ff911e85a57721d2c97c49ce71c1d2e28f3cb6b3aed3dfc9224b17c942"},"schema_version":"1.0","source":{"id":"1804.02532","kind":"arxiv","version":1}},"canonical_sha256":"1edc1414c8030769738d1805d660d60b7a61e66752a4f7de32685c364f10a5a5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1edc1414c8030769738d1805d660d60b7a61e66752a4f7de32685c364f10a5a5","first_computed_at":"2026-05-18T00:18:59.187942Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:18:59.187942Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ppBoJs8QkDaAEESZiZzD5mTioJUiBUndO6JfO+VPNnUtWH252+6aVC9vAZGIjxs6TuzCvlNQIRs0Ir7dzoEQAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:18:59.188358Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.02532","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:24024df5456ac6085db9562d87afb41cf8e07137bbbf7e0415d6f5d670e5b8ae","sha256:30dcbd43b4076ff0f4c61308084d36867c16df08076f4140ab1c6fe514ebccb4"],"state_sha256":"22d29559368d98a0f4a444f777883b236f9b87c125e84d15930ca155dce167c5"}