{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:6VHS2VS3IARTML2RYJIQPIOH35","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":"fcbbc6c497d4d959fb2862d2a96816e546ddc7e704aae0d076ed8f5471ff4c9d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-21T07:59:37Z","title_canon_sha256":"aa6a73b3b16313aa96f1adb835562a1a7bf95bbe2810ecbc6ce9b7ce05c19573"},"schema_version":"1.0","source":{"id":"1807.08104","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.08104","created_at":"2026-05-18T00:10:09Z"},{"alias_kind":"arxiv_version","alias_value":"1807.08104v1","created_at":"2026-05-18T00:10:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.08104","created_at":"2026-05-18T00:10:09Z"},{"alias_kind":"pith_short_12","alias_value":"6VHS2VS3IART","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"6VHS2VS3IARTML2R","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"6VHS2VS3","created_at":"2026-05-18T12:32:11Z"}],"graph_snapshots":[{"event_id":"sha256:372b9cde247f73c63eedbf096a511285294fffce3cab793c71150894f02de9ef","target":"graph","created_at":"2026-05-18T00:10:09Z","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 $k$-tuple total dominating set ($k$TDS) of a graph $G$ is a set $S$ of vertices in which every vertex in $G$ is adjacent to at least $k$ vertices in $S$. The minimum size of a $k$TDS is called the $k$-tuple total dominating number and it is denoted by $\\gamma_{\\times k,t}(G)$. We give a constructive proof of a general formula for $\\gamma_{\\times 3, t}(K_n \\Box K_m)$.","authors_text":"Behnaz Pahlavsay, Elisa Palezzato, Michele Torielli","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-21T07:59:37Z","title":"$3$-tuple total domination number of rook's graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.08104","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":"sha256:88a50955d23f0fb9227e956967ceb9bd0ffeeeace206c8a0161f2ae087918aab","target":"record","created_at":"2026-05-18T00:10:09Z","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":"fcbbc6c497d4d959fb2862d2a96816e546ddc7e704aae0d076ed8f5471ff4c9d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-21T07:59:37Z","title_canon_sha256":"aa6a73b3b16313aa96f1adb835562a1a7bf95bbe2810ecbc6ce9b7ce05c19573"},"schema_version":"1.0","source":{"id":"1807.08104","kind":"arxiv","version":1}},"canonical_sha256":"f54f2d565b4023362f51c25107a1c7df68a6835e4b79c338f209d359274c17fa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f54f2d565b4023362f51c25107a1c7df68a6835e4b79c338f209d359274c17fa","first_computed_at":"2026-05-18T00:10:09.298157Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:09.298157Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"E1kQDAXUuW3FvHblMCoqSZ+WGv/e/JcEXLD4ISxBRfXGjlSwiruOHRp2Cq+qVr9NA9j3i4gInKpmW5SXM19wDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:09.298839Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.08104","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:88a50955d23f0fb9227e956967ceb9bd0ffeeeace206c8a0161f2ae087918aab","sha256:372b9cde247f73c63eedbf096a511285294fffce3cab793c71150894f02de9ef"],"state_sha256":"b3500286ec97c6089adc751bbfd8c3edc9c008766d53750e2e27ecc385d27f7c"}