{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:CFJOPFFEZZOASRDAWZ4LLDM2GJ","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":"7118ac147a571fd0f996a4ed5a86c9f8ee03c5a1d22518b8a0909442fc65a394","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-07-07T08:21:43Z","title_canon_sha256":"a97e5e98d74c8e3bddda9b8327e9257dc4327e60ecc1a2878e053806941ef64b"},"schema_version":"1.0","source":{"id":"1607.01909","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.01909","created_at":"2026-05-18T00:53:50Z"},{"alias_kind":"arxiv_version","alias_value":"1607.01909v2","created_at":"2026-05-18T00:53:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.01909","created_at":"2026-05-18T00:53:50Z"},{"alias_kind":"pith_short_12","alias_value":"CFJOPFFEZZOA","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"CFJOPFFEZZOASRDA","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"CFJOPFFE","created_at":"2026-05-18T12:30:09Z"}],"graph_snapshots":[{"event_id":"sha256:a021a3b63c926f571392fabe175918c101d561ff04811fd05aaba0f597e3357d","target":"graph","created_at":"2026-05-18T00:53:50Z","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":"Ho proved in [A note on the total domination number, Util.Math. 77 (2008) 97--100] that the total domination number of the Cartesian product of any two graphs with no isolated vertices is at least one half of the product of their total domination numbers. We extend a result of Lu and Hou from [Total domination in the Cartesian product of a graph and $K_2$ or $C_n$, Util. Math. 83 (2010) 313--322] by characterizing the pairs of graphs $G$ and $H$ for which $\\gamma_t(G\\Box H)=\\frac{1}{2}\\gamma_t(G) \\gamma_t(H)\\,$, whenever $\\gamma_t(H)=2$. In addition, we present an infinite family of graphs $G_","authors_text":"Bo\\v{s}tjan Bre\\v{s}ar, Martin Milani\\v{c}, Tatiana Romina Hartinger, Tim Kos","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-07-07T08:21:43Z","title":"On total domination in the Cartesian product of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.01909","kind":"arxiv","version":2},"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":"sha256:42f26097cab984a040e6646357d7d618f24a83501fa42c458f9f9e77ab1c7921","target":"record","created_at":"2026-05-18T00:53:50Z","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":"7118ac147a571fd0f996a4ed5a86c9f8ee03c5a1d22518b8a0909442fc65a394","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-07-07T08:21:43Z","title_canon_sha256":"a97e5e98d74c8e3bddda9b8327e9257dc4327e60ecc1a2878e053806941ef64b"},"schema_version":"1.0","source":{"id":"1607.01909","kind":"arxiv","version":2}},"canonical_sha256":"1152e794a4ce5c094460b678b58d9a325dea89a8e4d84094a6884186a7297528","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1152e794a4ce5c094460b678b58d9a325dea89a8e4d84094a6884186a7297528","first_computed_at":"2026-05-18T00:53:50.829817Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:50.829817Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aGWwRJyMmmSXKNeNcWulWOTfOhcR3hRo0JYW9Qj7bVe0/LNUHt4iBos4Tc0fMfb7QwJoG+PpsrVgShKzMHJoBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:50.830201Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.01909","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:42f26097cab984a040e6646357d7d618f24a83501fa42c458f9f9e77ab1c7921","sha256:a021a3b63c926f571392fabe175918c101d561ff04811fd05aaba0f597e3357d"],"state_sha256":"ae686bc6d357dc29007f1b201db7ee1640c4999cfbddf2da78c6fd28b0316ec1"}