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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 $\\mathbb{M}$ "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1305.5692","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-24T11:32:59Z","cross_cats_sorted":[],"title_canon_sha256":"08370317043129f37938073e6c7d9bd9800bf0b3b0539ab68984cfe27e7f6c91","abstract_canon_sha256":"335c8e1b0b4bfa03e0398afc2873c6d8951ee8ac1b05092f542b36bcd2c83ce5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:03.492058Z","signature_b64":"L3gjrEYp/ucDyh/i65Jlq0qnj2PESQkDnPkZoipvisR18VaJO8zoY6rkkTCS0FWVOKqnnzaXbWrf+cAvLiJQCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"823e3d867c930dddea011122a041c228903b66a54b2b0dc5ff0e7d647424c033","last_reissued_at":"2026-05-18T03:25:03.491675Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:03.491675Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The bondage number of graphs on topological surfaces: degree-S vertices and the average degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Vladimir Samodivkin","submitted_at":"2013-05-24T11:32:59Z","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 $\\mathbb{M}$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.5692","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1305.5692","created_at":"2026-05-18T03:25:03.491724+00:00"},{"alias_kind":"arxiv_version","alias_value":"1305.5692v1","created_at":"2026-05-18T03:25:03.491724+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.5692","created_at":"2026-05-18T03:25:03.491724+00:00"},{"alias_kind":"pith_short_12","alias_value":"QI7D3BT4SMG5","created_at":"2026-05-18T12:27:57.521954+00:00"},{"alias_kind":"pith_short_16","alias_value":"QI7D3BT4SMG532QB","created_at":"2026-05-18T12:27:57.521954+00:00"},{"alias_kind":"pith_short_8","alias_value":"QI7D3BT4","created_at":"2026-05-18T12:27:57.521954+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QI7D3BT4SMG532QBCERKAQOCFC","json":"https://pith.science/pith/QI7D3BT4SMG532QBCERKAQOCFC.json","graph_json":"https://pith.science/api/pith-number/QI7D3BT4SMG532QBCERKAQOCFC/graph.json","events_json":"https://pith.science/api/pith-number/QI7D3BT4SMG532QBCERKAQOCFC/events.json","paper":"https://pith.science/paper/QI7D3BT4"},"agent_actions":{"view_html":"https://pith.science/pith/QI7D3BT4SMG532QBCERKAQOCFC","download_json":"https://pith.science/pith/QI7D3BT4SMG532QBCERKAQOCFC.json","view_paper":"https://pith.science/paper/QI7D3BT4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1305.5692&json=true","fetch_graph":"https://pith.science/api/pith-number/QI7D3BT4SMG532QBCERKAQOCFC/graph.json","fetch_events":"https://pith.science/api/pith-number/QI7D3BT4SMG532QBCERKAQOCFC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QI7D3BT4SMG532QBCERKAQOCFC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QI7D3BT4SMG532QBCERKAQOCFC/action/storage_attestation","attest_author":"https://pith.science/pith/QI7D3BT4SMG532QBCERKAQOCFC/action/author_attestation","sign_citation":"https://pith.science/pith/QI7D3BT4SMG532QBCERKAQOCFC/action/citation_signature","submit_replication":"https://pith.science/pith/QI7D3BT4SMG532QBCERKAQOCFC/action/replication_record"}},"created_at":"2026-05-18T03:25:03.491724+00:00","updated_at":"2026-05-18T03:25:03.491724+00:00"}