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For convenience, we use $G(n;D)$ to denote such a circulant graph.\n  A function $f:V(G)\\rightarrow\\mathbb{N}\\cup\\{0\\}$ is an integer $\\{k\\}$-domination function if for each $v\\in V(G)$, $\\sum_{u\\in N_G[v]}f(u)\\geq k.$ By considering all $\\{k\\}$-domination functions $f$, the minimum value of $\\sum_{v\\in V(G)}f(v)$ is "},"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":"1905.03388","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-05-08T23:14:56Z","cross_cats_sorted":[],"title_canon_sha256":"3fc8297cf19e2e1a2e9bcde97cfc5b0a7fdf00447ecc5acb05850eee0da8c4a0","abstract_canon_sha256":"052631aaca5d5ba9e89ca7019b5ab06534d38e81dd6650edd93f28f432048012"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:46:40.145639Z","signature_b64":"EZaP5ba+gvaf2SG8icTKKN/LGlzb8Dw5UrSQHKrkRYuNsCfUzQOw4UlqV0ng+4GZCelks9mlRTbjT3wQ556eAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"102ca32f05e22ba331128087c823e9c4537d810a3e5ffe7ead35c84c10c1f2a8","last_reissued_at":"2026-05-17T23:46:40.144528Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:46:40.144528Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the integer {k}-domination number of circulant graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chia-an Liu, Hung-Lin Fu, Yen-Jen Cheng","submitted_at":"2019-05-08T23:14:56Z","abstract_excerpt":"Let $G=(V,E)$ be a simple undirected graph. $G$ is a circulant graph defined on $V=\\mathbb{Z}_n$ with difference set $D\\subseteq \\{1,2,\\ldots,\\lfloor\\frac{n}{2}\\rfloor\\}$ provided two vertices $i$ and $j$ in $\\mathbb{Z}_n$ are adjacent if and only if $\\min\\{|i-j|, n-|i-j|\\}\\in D$. For convenience, we use $G(n;D)$ to denote such a circulant graph.\n  A function $f:V(G)\\rightarrow\\mathbb{N}\\cup\\{0\\}$ is an integer $\\{k\\}$-domination function if for each $v\\in V(G)$, $\\sum_{u\\in N_G[v]}f(u)\\geq k.$ By considering all $\\{k\\}$-domination functions $f$, the minimum value of $\\sum_{v\\in V(G)}f(v)$ is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.03388","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":"1905.03388","created_at":"2026-05-17T23:46:40.144648+00:00"},{"alias_kind":"arxiv_version","alias_value":"1905.03388v1","created_at":"2026-05-17T23:46:40.144648+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.03388","created_at":"2026-05-17T23:46:40.144648+00:00"},{"alias_kind":"pith_short_12","alias_value":"CAWKGLYF4IV2","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_16","alias_value":"CAWKGLYF4IV2GMIS","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_8","alias_value":"CAWKGLYF","created_at":"2026-05-18T12:33:12.712433+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/CAWKGLYF4IV2GMISQCD4QI7JYR","json":"https://pith.science/pith/CAWKGLYF4IV2GMISQCD4QI7JYR.json","graph_json":"https://pith.science/api/pith-number/CAWKGLYF4IV2GMISQCD4QI7JYR/graph.json","events_json":"https://pith.science/api/pith-number/CAWKGLYF4IV2GMISQCD4QI7JYR/events.json","paper":"https://pith.science/paper/CAWKGLYF"},"agent_actions":{"view_html":"https://pith.science/pith/CAWKGLYF4IV2GMISQCD4QI7JYR","download_json":"https://pith.science/pith/CAWKGLYF4IV2GMISQCD4QI7JYR.json","view_paper":"https://pith.science/paper/CAWKGLYF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1905.03388&json=true","fetch_graph":"https://pith.science/api/pith-number/CAWKGLYF4IV2GMISQCD4QI7JYR/graph.json","fetch_events":"https://pith.science/api/pith-number/CAWKGLYF4IV2GMISQCD4QI7JYR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CAWKGLYF4IV2GMISQCD4QI7JYR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CAWKGLYF4IV2GMISQCD4QI7JYR/action/storage_attestation","attest_author":"https://pith.science/pith/CAWKGLYF4IV2GMISQCD4QI7JYR/action/author_attestation","sign_citation":"https://pith.science/pith/CAWKGLYF4IV2GMISQCD4QI7JYR/action/citation_signature","submit_replication":"https://pith.science/pith/CAWKGLYF4IV2GMISQCD4QI7JYR/action/replication_record"}},"created_at":"2026-05-17T23:46:40.144648+00:00","updated_at":"2026-05-17T23:46:40.144648+00:00"}