{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:U4ZYCRS4ZYIGKAK4AMUAWSMEWB","short_pith_number":"pith:U4ZYCRS4","schema_version":"1.0","canonical_sha256":"a73381465cce1065015c03280b4984b07cbbbc0ff9734eb6db5a7942ec99d673","source":{"kind":"arxiv","id":"1608.04560","version":1},"attestation_state":"computed","paper":{"title":"Some Comments on the Slater number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dieter Rautenbach, Michael Gentner","submitted_at":"2016-08-16T11:54:29Z","abstract_excerpt":"Let $G$ be a graph with degree sequence $d_1\\geq \\ldots \\geq d_n$. Slater proposed $s\\ell(G)=\\min\\{ s: (d_1+1)+\\cdots+(d_s+1)\\geq n\\}$ as a lower bound on the domination number $\\gamma(G)$ of $G$. We show that deciding the equality of $\\gamma(G)$ and $s\\ell(G)$ for a given graph $G$ is NP-complete but that one can decide efficiently whether $\\gamma(G)>s\\ell(G)$ or $\\gamma(G)\\leq \\left(\\left\\lceil\\ln \\left(\\frac{n(G)}{s\\ell(G)}\\right)\\right\\rceil+1\\right)s\\ell(G)$. For real numbers $\\alpha$ and $\\beta$ with $\\alpha\\geq \\max\\{ 0,\\beta\\}$, let ${\\cal G}(\\alpha,\\beta)$ be the class of non-null gra"},"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":"1608.04560","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-08-16T11:54:29Z","cross_cats_sorted":[],"title_canon_sha256":"8be335c5519eef666eb576ea13bd24265baa74d776abbc6ab445950f4593c82d","abstract_canon_sha256":"c4a2096c9fa6189921753bd167e363ad222d85f3d4989e619c463564f1e1ff90"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:36.853084Z","signature_b64":"QK6yojibz5q9Kvf+FBgAsTwSRov4hAsJzVhcQph4jxYFjIWpJBfjTPGBcIj2xQJX9BqO7FM60qdYMovPRd7TBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a73381465cce1065015c03280b4984b07cbbbc0ff9734eb6db5a7942ec99d673","last_reissued_at":"2026-05-18T01:08:36.852444Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:36.852444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some Comments on the Slater number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dieter Rautenbach, Michael Gentner","submitted_at":"2016-08-16T11:54:29Z","abstract_excerpt":"Let $G$ be a graph with degree sequence $d_1\\geq \\ldots \\geq d_n$. Slater proposed $s\\ell(G)=\\min\\{ s: (d_1+1)+\\cdots+(d_s+1)\\geq n\\}$ as a lower bound on the domination number $\\gamma(G)$ of $G$. We show that deciding the equality of $\\gamma(G)$ and $s\\ell(G)$ for a given graph $G$ is NP-complete but that one can decide efficiently whether $\\gamma(G)>s\\ell(G)$ or $\\gamma(G)\\leq \\left(\\left\\lceil\\ln \\left(\\frac{n(G)}{s\\ell(G)}\\right)\\right\\rceil+1\\right)s\\ell(G)$. For real numbers $\\alpha$ and $\\beta$ with $\\alpha\\geq \\max\\{ 0,\\beta\\}$, let ${\\cal G}(\\alpha,\\beta)$ be the class of non-null gra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.04560","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":"1608.04560","created_at":"2026-05-18T01:08:36.852549+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.04560v1","created_at":"2026-05-18T01:08:36.852549+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.04560","created_at":"2026-05-18T01:08:36.852549+00:00"},{"alias_kind":"pith_short_12","alias_value":"U4ZYCRS4ZYIG","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_16","alias_value":"U4ZYCRS4ZYIGKAK4","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_8","alias_value":"U4ZYCRS4","created_at":"2026-05-18T12:30:46.583412+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/U4ZYCRS4ZYIGKAK4AMUAWSMEWB","json":"https://pith.science/pith/U4ZYCRS4ZYIGKAK4AMUAWSMEWB.json","graph_json":"https://pith.science/api/pith-number/U4ZYCRS4ZYIGKAK4AMUAWSMEWB/graph.json","events_json":"https://pith.science/api/pith-number/U4ZYCRS4ZYIGKAK4AMUAWSMEWB/events.json","paper":"https://pith.science/paper/U4ZYCRS4"},"agent_actions":{"view_html":"https://pith.science/pith/U4ZYCRS4ZYIGKAK4AMUAWSMEWB","download_json":"https://pith.science/pith/U4ZYCRS4ZYIGKAK4AMUAWSMEWB.json","view_paper":"https://pith.science/paper/U4ZYCRS4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.04560&json=true","fetch_graph":"https://pith.science/api/pith-number/U4ZYCRS4ZYIGKAK4AMUAWSMEWB/graph.json","fetch_events":"https://pith.science/api/pith-number/U4ZYCRS4ZYIGKAK4AMUAWSMEWB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/U4ZYCRS4ZYIGKAK4AMUAWSMEWB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/U4ZYCRS4ZYIGKAK4AMUAWSMEWB/action/storage_attestation","attest_author":"https://pith.science/pith/U4ZYCRS4ZYIGKAK4AMUAWSMEWB/action/author_attestation","sign_citation":"https://pith.science/pith/U4ZYCRS4ZYIGKAK4AMUAWSMEWB/action/citation_signature","submit_replication":"https://pith.science/pith/U4ZYCRS4ZYIGKAK4AMUAWSMEWB/action/replication_record"}},"created_at":"2026-05-18T01:08:36.852549+00:00","updated_at":"2026-05-18T01:08:36.852549+00:00"}