{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:ECRPOCDPTO2VH7N466KR5DTGC2","short_pith_number":"pith:ECRPOCDP","schema_version":"1.0","canonical_sha256":"20a2f7086f9bb553fdbcf7951e8e661690551beb14270d81f10928bf455d10c0","source":{"kind":"arxiv","id":"1801.04965","version":1},"attestation_state":"computed","paper":{"title":"Changing and unchanging of the domination number of a graph: Path addition numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Vladimir Samodivkin","submitted_at":"2018-01-15T19:51:26Z","abstract_excerpt":"Given a graph $G = (V,E)$ and two its distinct vertices $u$ and $v$. The $(u,v)$-$P_k$-{\\em addition graph} of $G$ is the graph $G_{u,v,k-2}$ obtained from disjoint union of $G$ and a path $P_k: x_0,x_1,..,x_{k-1}$, $k \\geq 2$, by identifying the vertices $u$ and $x_0$, and identifying the vertices $v$ and $x_{k-1}$. We prove that (a) $ \\gamma(G)-1 \\leq \\gamma(G_{u,v,k})$ for all $k \\geq 1$, and (b) $\\gamma(G_{u,v,k}) > \\gamma(G)$ when $k \\geq 5$. We also provide necessary and sufficient conditions for the equality $\\gamma(G_{u,v,k}) = \\gamma(G)$ to be valid for each pair $u,v \\in V(G)$. pair "},"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":"1801.04965","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-15T19:51:26Z","cross_cats_sorted":[],"title_canon_sha256":"6ab3a65a851c5b41151e08bd92f8c961120c45d81d0ea8a983da7039a1b53bcc","abstract_canon_sha256":"1ed5d5ebb182ccf1d7c0eefd64c6c0a8b95ee0a0a7711d06f9bd7f54dd9cb043"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:25:50.468628Z","signature_b64":"C9o0mtAkrwJ4kb91MmG/QglKJjv+chsc4C1N3GjuwUjDLcMsc6/D8A5mse3c+cQCLRsWzlHXoJOGj25Wxk20Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"20a2f7086f9bb553fdbcf7951e8e661690551beb14270d81f10928bf455d10c0","last_reissued_at":"2026-05-18T00:25:50.468009Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:25:50.468009Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Changing and unchanging of the domination number of a graph: Path addition numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Vladimir Samodivkin","submitted_at":"2018-01-15T19:51:26Z","abstract_excerpt":"Given a graph $G = (V,E)$ and two its distinct vertices $u$ and $v$. The $(u,v)$-$P_k$-{\\em addition graph} of $G$ is the graph $G_{u,v,k-2}$ obtained from disjoint union of $G$ and a path $P_k: x_0,x_1,..,x_{k-1}$, $k \\geq 2$, by identifying the vertices $u$ and $x_0$, and identifying the vertices $v$ and $x_{k-1}$. We prove that (a) $ \\gamma(G)-1 \\leq \\gamma(G_{u,v,k})$ for all $k \\geq 1$, and (b) $\\gamma(G_{u,v,k}) > \\gamma(G)$ when $k \\geq 5$. We also provide necessary and sufficient conditions for the equality $\\gamma(G_{u,v,k}) = \\gamma(G)$ to be valid for each pair $u,v \\in V(G)$. pair "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.04965","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":"1801.04965","created_at":"2026-05-18T00:25:50.468090+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.04965v1","created_at":"2026-05-18T00:25:50.468090+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.04965","created_at":"2026-05-18T00:25:50.468090+00:00"},{"alias_kind":"pith_short_12","alias_value":"ECRPOCDPTO2V","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_16","alias_value":"ECRPOCDPTO2VH7N4","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_8","alias_value":"ECRPOCDP","created_at":"2026-05-18T12:32:22.470017+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/ECRPOCDPTO2VH7N466KR5DTGC2","json":"https://pith.science/pith/ECRPOCDPTO2VH7N466KR5DTGC2.json","graph_json":"https://pith.science/api/pith-number/ECRPOCDPTO2VH7N466KR5DTGC2/graph.json","events_json":"https://pith.science/api/pith-number/ECRPOCDPTO2VH7N466KR5DTGC2/events.json","paper":"https://pith.science/paper/ECRPOCDP"},"agent_actions":{"view_html":"https://pith.science/pith/ECRPOCDPTO2VH7N466KR5DTGC2","download_json":"https://pith.science/pith/ECRPOCDPTO2VH7N466KR5DTGC2.json","view_paper":"https://pith.science/paper/ECRPOCDP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.04965&json=true","fetch_graph":"https://pith.science/api/pith-number/ECRPOCDPTO2VH7N466KR5DTGC2/graph.json","fetch_events":"https://pith.science/api/pith-number/ECRPOCDPTO2VH7N466KR5DTGC2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ECRPOCDPTO2VH7N466KR5DTGC2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ECRPOCDPTO2VH7N466KR5DTGC2/action/storage_attestation","attest_author":"https://pith.science/pith/ECRPOCDPTO2VH7N466KR5DTGC2/action/author_attestation","sign_citation":"https://pith.science/pith/ECRPOCDPTO2VH7N466KR5DTGC2/action/citation_signature","submit_replication":"https://pith.science/pith/ECRPOCDPTO2VH7N466KR5DTGC2/action/replication_record"}},"created_at":"2026-05-18T00:25:50.468090+00:00","updated_at":"2026-05-18T00:25:50.468090+00:00"}