{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:LHADXKWP6X3JBN466CCU27SVPA","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":"f2159d3e667f4f4a23daa2b113a63dc1d0326fc54baa9b710137f0f8677d1fad","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-07T12:22:07Z","title_canon_sha256":"765bd7d824e7386fcd8db22a3cff920b48273b20aae8c1ed866db2a0775da000"},"schema_version":"1.0","source":{"id":"1804.02550","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.02550","created_at":"2026-05-18T00:18:59Z"},{"alias_kind":"arxiv_version","alias_value":"1804.02550v1","created_at":"2026-05-18T00:18:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.02550","created_at":"2026-05-18T00:18:59Z"},{"alias_kind":"pith_short_12","alias_value":"LHADXKWP6X3J","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_16","alias_value":"LHADXKWP6X3JBN46","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_8","alias_value":"LHADXKWP","created_at":"2026-05-18T12:32:37Z"}],"graph_snapshots":[{"event_id":"sha256:06363ad77aaad0020618f04b0dc9d7adf1eef6a29814d0944ab3e152ac2c0fa0","target":"graph","created_at":"2026-05-18T00:18:59Z","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":"A subset $D$ of vertices of a graph $G$ is a dominating set if for each $u\\in V(G)\\setminus D$, $u$ is adjacent to some vertex $v\\in D$. The domination number, $\\gamma(G)$ of $G$, is the minimum cardinality of a dominating set of $G$. For an even integer $n\\ge2$ and $1\\le\\Delta\\le\\lfloor\\log_2n\\rfloor$, a Kn\\\"odel graph $W_{\\Delta,n}$ is a $Delta$-regular bipartite graph of even order $n$, with vertices$(i,j)$, for $i=1,2$ and $0\\le j\\le n/2-1$, where for every $j$, $0\\le j\\le n/2-1$, there is an edge between $(1,j)$ and $(2,j+2^k-1 \\text{(mod(n/2)})$, for $k=0,1,\\cdots,\\Delta-1$. In this pape","authors_text":"Doost Ali Mojdeh, Esmaeil Nazari, Seyed Reza Musawi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-07T12:22:07Z","title":"Domination in 4-regular Kn\\\"odel graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.02550","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:c5f462869020992aba66d2e4360156e28262affdb08d9f14bfa7d13d1691848a","target":"record","created_at":"2026-05-18T00:18:59Z","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":"f2159d3e667f4f4a23daa2b113a63dc1d0326fc54baa9b710137f0f8677d1fad","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-07T12:22:07Z","title_canon_sha256":"765bd7d824e7386fcd8db22a3cff920b48273b20aae8c1ed866db2a0775da000"},"schema_version":"1.0","source":{"id":"1804.02550","kind":"arxiv","version":1}},"canonical_sha256":"59c03baacff5f690b79ef0854d7e55783d8c3b5a49a120cd09ba919cc3f33d05","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"59c03baacff5f690b79ef0854d7e55783d8c3b5a49a120cd09ba919cc3f33d05","first_computed_at":"2026-05-18T00:18:59.149286Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:18:59.149286Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2HRW71ZO29sixs9VMW/llJQD9IXdAZL+pA7Ry8FiaNxGDRaGcCYWnJjiDv0wDu1vsFswGz1AroNHuC+FMrcUDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:18:59.149766Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.02550","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c5f462869020992aba66d2e4360156e28262affdb08d9f14bfa7d13d1691848a","sha256:06363ad77aaad0020618f04b0dc9d7adf1eef6a29814d0944ab3e152ac2c0fa0"],"state_sha256":"a5de835f3b0b19b6d4c523ec80dcc4d0b628d01ff696d2ad76cb168448c0e3d1"}