{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:V4OKPPFWI32HVQXKE4NOTSEW4V","short_pith_number":"pith:V4OKPPFW","canonical_record":{"source":{"id":"1805.03335","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-09T00:58:46Z","cross_cats_sorted":[],"title_canon_sha256":"30b3173579113f95a7447f0414eccbcc8c451998ec2898ff5068ca13f801d136","abstract_canon_sha256":"b093de00589f2c4a35d49879f31b9df792bca6402c937be29590778fe85aaa25"},"schema_version":"1.0"},"canonical_sha256":"af1ca7bcb646f47ac2ea271ae9c896e54e4443f4914f73ed46c940bd0d99bcfb","source":{"kind":"arxiv","id":"1805.03335","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.03335","created_at":"2026-05-18T00:16:20Z"},{"alias_kind":"arxiv_version","alias_value":"1805.03335v1","created_at":"2026-05-18T00:16:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.03335","created_at":"2026-05-18T00:16:20Z"},{"alias_kind":"pith_short_12","alias_value":"V4OKPPFWI32H","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"V4OKPPFWI32HVQXK","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"V4OKPPFW","created_at":"2026-05-18T12:32:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:V4OKPPFWI32HVQXKE4NOTSEW4V","target":"record","payload":{"canonical_record":{"source":{"id":"1805.03335","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-09T00:58:46Z","cross_cats_sorted":[],"title_canon_sha256":"30b3173579113f95a7447f0414eccbcc8c451998ec2898ff5068ca13f801d136","abstract_canon_sha256":"b093de00589f2c4a35d49879f31b9df792bca6402c937be29590778fe85aaa25"},"schema_version":"1.0"},"canonical_sha256":"af1ca7bcb646f47ac2ea271ae9c896e54e4443f4914f73ed46c940bd0d99bcfb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:20.466455Z","signature_b64":"3b765+PydnowSx0yWan7OvBRguUbt64srlKfj0VQ49Z2stctQPQ7E81QWraU3Xe6WRgK7WXm39A6V2QJQbSvAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"af1ca7bcb646f47ac2ea271ae9c896e54e4443f4914f73ed46c940bd0d99bcfb","last_reissued_at":"2026-05-18T00:16:20.465821Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:20.465821Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.03335","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:16:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fybZNKKUY1rjjbqlJBJm1hgN0vmtb7ITRVgRESNRl5NZnj3wJ5L0RtdVLedk73C0sPM5DQc++awMoiZUAx44AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T06:58:09.975107Z"},"content_sha256":"efe96a32bae260bc22ad24206828c9c4fa2dba28ded467e2e531df49580b11e5","schema_version":"1.0","event_id":"sha256:efe96a32bae260bc22ad24206828c9c4fa2dba28ded467e2e531df49580b11e5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:V4OKPPFWI32HVQXKE4NOTSEW4V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Perfect Domination in Knights Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Renu Laskar, Soumendra Ganguly, Todd Fenstermacher","submitted_at":"2018-05-09T00:58:46Z","abstract_excerpt":"For a graph $G = (V,E),$ a subset $S$ of $V$ is a perfect dominating set of $G$ if every vertex not in $S$ is adjacent to exactly one vertex in $S.$ The perfect domination number, $\\gamma_p(G),$ is the minimum cardinality of a perfect dominating set of $G.$ The perfect domination number is found for knights graphs on square, rectangular, and infinite chessboards. Indeed, exact values or bounds are given for all chessboards except those with 3 rows and number of columns congruent to 1, 2, or 3 modulo 8."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.03335","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:16:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AbwYRqR7uBq47fSEp5qN2AEeiYqgniXYqn6Ue41zEjNU2mSJrsnbocDsutnBWhR9ljQHD/II5BjLrVvef90iAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T06:58:09.975492Z"},"content_sha256":"d3d319152c6530b1728bad59988ae3817e2f9b776ea8b824b07d9deea97ffc08","schema_version":"1.0","event_id":"sha256:d3d319152c6530b1728bad59988ae3817e2f9b776ea8b824b07d9deea97ffc08"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V4OKPPFWI32HVQXKE4NOTSEW4V/bundle.json","state_url":"https://pith.science/pith/V4OKPPFWI32HVQXKE4NOTSEW4V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V4OKPPFWI32HVQXKE4NOTSEW4V/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-02T06:58:09Z","links":{"resolver":"https://pith.science/pith/V4OKPPFWI32HVQXKE4NOTSEW4V","bundle":"https://pith.science/pith/V4OKPPFWI32HVQXKE4NOTSEW4V/bundle.json","state":"https://pith.science/pith/V4OKPPFWI32HVQXKE4NOTSEW4V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V4OKPPFWI32HVQXKE4NOTSEW4V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:V4OKPPFWI32HVQXKE4NOTSEW4V","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":"b093de00589f2c4a35d49879f31b9df792bca6402c937be29590778fe85aaa25","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-09T00:58:46Z","title_canon_sha256":"30b3173579113f95a7447f0414eccbcc8c451998ec2898ff5068ca13f801d136"},"schema_version":"1.0","source":{"id":"1805.03335","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.03335","created_at":"2026-05-18T00:16:20Z"},{"alias_kind":"arxiv_version","alias_value":"1805.03335v1","created_at":"2026-05-18T00:16:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.03335","created_at":"2026-05-18T00:16:20Z"},{"alias_kind":"pith_short_12","alias_value":"V4OKPPFWI32H","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"V4OKPPFWI32HVQXK","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"V4OKPPFW","created_at":"2026-05-18T12:32:56Z"}],"graph_snapshots":[{"event_id":"sha256:d3d319152c6530b1728bad59988ae3817e2f9b776ea8b824b07d9deea97ffc08","target":"graph","created_at":"2026-05-18T00:16:20Z","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":"For a graph $G = (V,E),$ a subset $S$ of $V$ is a perfect dominating set of $G$ if every vertex not in $S$ is adjacent to exactly one vertex in $S.$ The perfect domination number, $\\gamma_p(G),$ is the minimum cardinality of a perfect dominating set of $G.$ The perfect domination number is found for knights graphs on square, rectangular, and infinite chessboards. Indeed, exact values or bounds are given for all chessboards except those with 3 rows and number of columns congruent to 1, 2, or 3 modulo 8.","authors_text":"Renu Laskar, Soumendra Ganguly, Todd Fenstermacher","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-09T00:58:46Z","title":"Perfect Domination in Knights Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.03335","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:efe96a32bae260bc22ad24206828c9c4fa2dba28ded467e2e531df49580b11e5","target":"record","created_at":"2026-05-18T00:16:20Z","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":"b093de00589f2c4a35d49879f31b9df792bca6402c937be29590778fe85aaa25","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-09T00:58:46Z","title_canon_sha256":"30b3173579113f95a7447f0414eccbcc8c451998ec2898ff5068ca13f801d136"},"schema_version":"1.0","source":{"id":"1805.03335","kind":"arxiv","version":1}},"canonical_sha256":"af1ca7bcb646f47ac2ea271ae9c896e54e4443f4914f73ed46c940bd0d99bcfb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"af1ca7bcb646f47ac2ea271ae9c896e54e4443f4914f73ed46c940bd0d99bcfb","first_computed_at":"2026-05-18T00:16:20.465821Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:16:20.465821Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3b765+PydnowSx0yWan7OvBRguUbt64srlKfj0VQ49Z2stctQPQ7E81QWraU3Xe6WRgK7WXm39A6V2QJQbSvAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:16:20.466455Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.03335","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:efe96a32bae260bc22ad24206828c9c4fa2dba28ded467e2e531df49580b11e5","sha256:d3d319152c6530b1728bad59988ae3817e2f9b776ea8b824b07d9deea97ffc08"],"state_sha256":"989033add4edb5df733cb921ba0ac216fad2371cdae0401ac97013d5cf952a4b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zvFuxSN/y9Dw4o6JwFXYJntiMLN83zoogiMuRS8sQN/cyCnUawMmfvtcdQZddoMegeXUlpiCRZAw63AoXGsRBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T06:58:09.977457Z","bundle_sha256":"6c2228ed1a7f8cfb9862bbb1ae9112c524ffbb423237e0ec50bcc12a52f92c2c"}}