{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:73QEOD3N5DDQYPMF7BILSBQ55K","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":"08652c77df9bc859f6b053d52ad4ad665d66151b44f0fb00e3571d90f6ee47ef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2007-11-27T20:40:28Z","title_canon_sha256":"05255466e20afaf50ef74b2a1597842fb2895b97626ec2df3ea5cfe47cb13699"},"schema_version":"1.0","source":{"id":"0711.4343","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0711.4343","created_at":"2026-05-18T02:24:57Z"},{"alias_kind":"arxiv_version","alias_value":"0711.4343v4","created_at":"2026-05-18T02:24:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0711.4343","created_at":"2026-05-18T02:24:57Z"},{"alias_kind":"pith_short_12","alias_value":"73QEOD3N5DDQ","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"73QEOD3N5DDQYPMF","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"73QEOD3N","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:f061825585beed6e73cd10fb4102cd6195be2f860a07e4145b4cfb194650ece5","target":"graph","created_at":"2026-05-18T02:24:57Z","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":"We show there is an uncountable number of parallel total perfect codes in the integer lattice graph ${\\Lambda}$ of $\\R^2$. In contrast, there is just one 1-perfect code in ${\\Lambda}$ and one total perfect code in ${\\Lambda}$ restricting to total perfect codes of rectangular grid graphs (yielding an asymmetric, Penrose, tiling of the plane). We characterize all cycle products $C_m\\times C_n$ with parallel total perfect codes, and the $d$-perfect and total perfect code partitions of ${\\Lambda}$ and $C_m\\times C_n$, the former having as quotient graph the undirected Cayley graphs of $\\Z_{2d^2+2d","authors_text":"Italo J. Dejter","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2007-11-27T20:40:28Z","title":"Perfect domination in regular grid graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0711.4343","kind":"arxiv","version":4},"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:ba2019abd17404c93d362d87246e29c9e7907543f2edae02c9592f7fe09147c1","target":"record","created_at":"2026-05-18T02:24:57Z","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":"08652c77df9bc859f6b053d52ad4ad665d66151b44f0fb00e3571d90f6ee47ef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2007-11-27T20:40:28Z","title_canon_sha256":"05255466e20afaf50ef74b2a1597842fb2895b97626ec2df3ea5cfe47cb13699"},"schema_version":"1.0","source":{"id":"0711.4343","kind":"arxiv","version":4}},"canonical_sha256":"fee0470f6de8c70c3d85f850b9061deaab7c3ff964efc7121f42934fab7a29e5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fee0470f6de8c70c3d85f850b9061deaab7c3ff964efc7121f42934fab7a29e5","first_computed_at":"2026-05-18T02:24:57.753744Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:24:57.753744Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FWdTh3czQWtPqgYosvccLFB0jwbSfk5FF6z8TbnyKYt/MiMFyqQ9jHH9gbNPKj4Aptbq6GQsmDI9EprLzXt7Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:24:57.754658Z","signed_message":"canonical_sha256_bytes"},"source_id":"0711.4343","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ba2019abd17404c93d362d87246e29c9e7907543f2edae02c9592f7fe09147c1","sha256:f061825585beed6e73cd10fb4102cd6195be2f860a07e4145b4cfb194650ece5"],"state_sha256":"0a59e698a2502d5fd9aef3eb7a620d088ab189512a9ebc90409a107f86e1caf1"}