{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:QI7D3BT4SMG532QBCERKAQOCFC","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":"335c8e1b0b4bfa03e0398afc2873c6d8951ee8ac1b05092f542b36bcd2c83ce5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-24T11:32:59Z","title_canon_sha256":"08370317043129f37938073e6c7d9bd9800bf0b3b0539ab68984cfe27e7f6c91"},"schema_version":"1.0","source":{"id":"1305.5692","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.5692","created_at":"2026-05-18T03:25:03Z"},{"alias_kind":"arxiv_version","alias_value":"1305.5692v1","created_at":"2026-05-18T03:25:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.5692","created_at":"2026-05-18T03:25:03Z"},{"alias_kind":"pith_short_12","alias_value":"QI7D3BT4SMG5","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"QI7D3BT4SMG532QB","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"QI7D3BT4","created_at":"2026-05-18T12:27:57Z"}],"graph_snapshots":[{"event_id":"sha256:82c412c15966820ee8b0167bdcf64305cf539372f1f6b586fab2236b02e3748a","target":"graph","created_at":"2026-05-18T03:25:03Z","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":"The bondage number $b(G)$ of a graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with larger domination number. An orientable surface $\\mathbb{S}_h$ of genus $h$, $h \\geq 0$, is obtained from the sphere $\\mathbb{S}_0$ by adding $h$ handles. A non-orientable surface $\\mathbb{N}_q$ of genus $q$, $q \\geq 1$, is obtained from the sphere by adding $q$ crosscaps. The Euler characteristic of a surface is defined by $\\chi(\\mathbb{S}_h) = 2 - 2h$ and $\\chi(\\mathbb{S}_q)= 2-q$. Let $G$ be a connected graph of order $n$ which is 2-cell embedded on a surface $\\mathbb{M}$ ","authors_text":"Vladimir Samodivkin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-24T11:32:59Z","title":"The bondage number of graphs on topological surfaces: degree-S vertices and the average degree"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.5692","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:ee21375305951559a43022380a643d4c9bb6eb52401771c75248f7f656e13ecc","target":"record","created_at":"2026-05-18T03:25:03Z","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":"335c8e1b0b4bfa03e0398afc2873c6d8951ee8ac1b05092f542b36bcd2c83ce5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-24T11:32:59Z","title_canon_sha256":"08370317043129f37938073e6c7d9bd9800bf0b3b0539ab68984cfe27e7f6c91"},"schema_version":"1.0","source":{"id":"1305.5692","kind":"arxiv","version":1}},"canonical_sha256":"823e3d867c930dddea011122a041c228903b66a54b2b0dc5ff0e7d647424c033","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"823e3d867c930dddea011122a041c228903b66a54b2b0dc5ff0e7d647424c033","first_computed_at":"2026-05-18T03:25:03.491675Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:25:03.491675Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"L3gjrEYp/ucDyh/i65Jlq0qnj2PESQkDnPkZoipvisR18VaJO8zoY6rkkTCS0FWVOKqnnzaXbWrf+cAvLiJQCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:25:03.492058Z","signed_message":"canonical_sha256_bytes"},"source_id":"1305.5692","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ee21375305951559a43022380a643d4c9bb6eb52401771c75248f7f656e13ecc","sha256:82c412c15966820ee8b0167bdcf64305cf539372f1f6b586fab2236b02e3748a"],"state_sha256":"938529259c870224c2465de8f689fb629b505ea7007520436267728c39e971a1"}