{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:VMOMOVL4R4AOMHAKOLKAX66IRT","short_pith_number":"pith:VMOMOVL4","canonical_record":{"source":{"id":"1712.08780","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-23T14:42:14Z","cross_cats_sorted":[],"title_canon_sha256":"177a6770bd8380876f0c3d542ecb8449f07d926e09c6a013d4d13299d6d4ab18","abstract_canon_sha256":"ffd7207dfa3f36bc54068668dc4d125ee617b7bd19d0927ac0a9098e32e4a3e1"},"schema_version":"1.0"},"canonical_sha256":"ab1cc7557c8f00e61c0a72d40bfbc88cdb974779a8ffd3ac866f9950ec51534f","source":{"kind":"arxiv","id":"1712.08780","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.08780","created_at":"2026-05-18T00:27:16Z"},{"alias_kind":"arxiv_version","alias_value":"1712.08780v1","created_at":"2026-05-18T00:27:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.08780","created_at":"2026-05-18T00:27:16Z"},{"alias_kind":"pith_short_12","alias_value":"VMOMOVL4R4AO","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"VMOMOVL4R4AOMHAK","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"VMOMOVL4","created_at":"2026-05-18T12:31:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:VMOMOVL4R4AOMHAKOLKAX66IRT","target":"record","payload":{"canonical_record":{"source":{"id":"1712.08780","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-23T14:42:14Z","cross_cats_sorted":[],"title_canon_sha256":"177a6770bd8380876f0c3d542ecb8449f07d926e09c6a013d4d13299d6d4ab18","abstract_canon_sha256":"ffd7207dfa3f36bc54068668dc4d125ee617b7bd19d0927ac0a9098e32e4a3e1"},"schema_version":"1.0"},"canonical_sha256":"ab1cc7557c8f00e61c0a72d40bfbc88cdb974779a8ffd3ac866f9950ec51534f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:27:16.837125Z","signature_b64":"pqNb/Aiam8n2gi+tOEjoh3CYcYp4fFY3txssXbeIJ+9WNTZN0oq64av1qWJTjjprfoEwUFu7mOiLyYmn2y/oDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab1cc7557c8f00e61c0a72d40bfbc88cdb974779a8ffd3ac866f9950ec51534f","last_reissued_at":"2026-05-18T00:27:16.836692Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:27:16.836692Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1712.08780","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":"2026-05-18T00:27:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OP3h9DvvL3WNbEOz+XryZ6FWAlQvke83Jj7I5moVAVJO40PU1o4/q8/Wc+//R8J/l8dz8EZ2Y2xnHHWLizgODw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T05:12:26.633207Z"},"content_sha256":"2bb88364b93904bc41a769f8f47b9ebdbaf0b5afe26691824f911907ee809fa9","schema_version":"1.0","event_id":"sha256:2bb88364b93904bc41a769f8f47b9ebdbaf0b5afe26691824f911907ee809fa9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:VMOMOVL4R4AOMHAKOLKAX66IRT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Grundy total domination number in product graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bal\\'azs Patk\\'os, Bo\\v{s}tjan Bre\\v{s}ar, Csilla Bujt\\'as, Ga\\v{s}per Ko\\v{s}mrlj, M\\'at\\'e Vizer, Sandi Klav\\v{z}ar, Tanja Gologranc, Tilen Marc, Zsolt Tuza","submitted_at":"2017-12-23T14:42:14Z","abstract_excerpt":"A longest sequence $(v_1,\\ldots,v_k)$ of vertices of a graph $G$ is a Grundy total dominating sequence of $G$ if for all $i$, $N(v_i) \\setminus \\bigcup_{j=1}^{i-1}N(v_j)\\not=\\emptyset$. The length $k$ of the sequence is called the Grundy total domination number of $G$ and denoted $\\gamma_{gr}^{t}(G)$. In this paper, the Grundy total domination number is studied on four standard graph products. For the direct product we show that $\\gamma_{gr}^t(G\\times H) \\geq \\gamma_{gr}^t(G)\\gamma_{gr}^t(H)$, conjecture that the equality always holds, and prove the conjecture in several special cases. For the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.08780","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":"2026-05-18T00:27:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yi8wpS6AM5oN7lrXUYCV20GUCY3WW6bvQ1GmEkeRrWZJ0H8gwAmJzp7QMChubCkSw+e2XvpBmIrI9QQyKL13Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T05:12:26.633556Z"},"content_sha256":"624593c4672f32eb3437477cabdd540b191d3b6d29207121d39c61ed408f89a3","schema_version":"1.0","event_id":"sha256:624593c4672f32eb3437477cabdd540b191d3b6d29207121d39c61ed408f89a3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VMOMOVL4R4AOMHAKOLKAX66IRT/bundle.json","state_url":"https://pith.science/pith/VMOMOVL4R4AOMHAKOLKAX66IRT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VMOMOVL4R4AOMHAKOLKAX66IRT/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":"2026-06-20T05:12:26Z","links":{"resolver":"https://pith.science/pith/VMOMOVL4R4AOMHAKOLKAX66IRT","bundle":"https://pith.science/pith/VMOMOVL4R4AOMHAKOLKAX66IRT/bundle.json","state":"https://pith.science/pith/VMOMOVL4R4AOMHAKOLKAX66IRT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VMOMOVL4R4AOMHAKOLKAX66IRT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:VMOMOVL4R4AOMHAKOLKAX66IRT","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":"ffd7207dfa3f36bc54068668dc4d125ee617b7bd19d0927ac0a9098e32e4a3e1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-23T14:42:14Z","title_canon_sha256":"177a6770bd8380876f0c3d542ecb8449f07d926e09c6a013d4d13299d6d4ab18"},"schema_version":"1.0","source":{"id":"1712.08780","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.08780","created_at":"2026-05-18T00:27:16Z"},{"alias_kind":"arxiv_version","alias_value":"1712.08780v1","created_at":"2026-05-18T00:27:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.08780","created_at":"2026-05-18T00:27:16Z"},{"alias_kind":"pith_short_12","alias_value":"VMOMOVL4R4AO","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"VMOMOVL4R4AOMHAK","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"VMOMOVL4","created_at":"2026-05-18T12:31:49Z"}],"graph_snapshots":[{"event_id":"sha256:624593c4672f32eb3437477cabdd540b191d3b6d29207121d39c61ed408f89a3","target":"graph","created_at":"2026-05-18T00:27:16Z","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 longest sequence $(v_1,\\ldots,v_k)$ of vertices of a graph $G$ is a Grundy total dominating sequence of $G$ if for all $i$, $N(v_i) \\setminus \\bigcup_{j=1}^{i-1}N(v_j)\\not=\\emptyset$. The length $k$ of the sequence is called the Grundy total domination number of $G$ and denoted $\\gamma_{gr}^{t}(G)$. In this paper, the Grundy total domination number is studied on four standard graph products. For the direct product we show that $\\gamma_{gr}^t(G\\times H) \\geq \\gamma_{gr}^t(G)\\gamma_{gr}^t(H)$, conjecture that the equality always holds, and prove the conjecture in several special cases. For the","authors_text":"Bal\\'azs Patk\\'os, Bo\\v{s}tjan Bre\\v{s}ar, Csilla Bujt\\'as, Ga\\v{s}per Ko\\v{s}mrlj, M\\'at\\'e Vizer, Sandi Klav\\v{z}ar, Tanja Gologranc, Tilen Marc, Zsolt Tuza","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-23T14:42:14Z","title":"On Grundy total domination number in product graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.08780","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":"sha256:2bb88364b93904bc41a769f8f47b9ebdbaf0b5afe26691824f911907ee809fa9","target":"record","created_at":"2026-05-18T00:27:16Z","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":"ffd7207dfa3f36bc54068668dc4d125ee617b7bd19d0927ac0a9098e32e4a3e1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-23T14:42:14Z","title_canon_sha256":"177a6770bd8380876f0c3d542ecb8449f07d926e09c6a013d4d13299d6d4ab18"},"schema_version":"1.0","source":{"id":"1712.08780","kind":"arxiv","version":1}},"canonical_sha256":"ab1cc7557c8f00e61c0a72d40bfbc88cdb974779a8ffd3ac866f9950ec51534f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ab1cc7557c8f00e61c0a72d40bfbc88cdb974779a8ffd3ac866f9950ec51534f","first_computed_at":"2026-05-18T00:27:16.836692Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:27:16.836692Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pqNb/Aiam8n2gi+tOEjoh3CYcYp4fFY3txssXbeIJ+9WNTZN0oq64av1qWJTjjprfoEwUFu7mOiLyYmn2y/oDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:27:16.837125Z","signed_message":"canonical_sha256_bytes"},"source_id":"1712.08780","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2bb88364b93904bc41a769f8f47b9ebdbaf0b5afe26691824f911907ee809fa9","sha256:624593c4672f32eb3437477cabdd540b191d3b6d29207121d39c61ed408f89a3"],"state_sha256":"f3098b1fe9f6d7c807ffe6aece02e03b121d55cdb1d3b7c0f4557d9a84718a6d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AbM6XCkpuT94oDlfpJCKV+gTYXIuNOlLliMudFOiMMt/lDdJqtwM3pJI7a9r/e+cok5q2zwagKt5T8ji27wOBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T05:12:26.635360Z","bundle_sha256":"bdf882866932963cba69f10a7771d9ba95b86ad29743d9418b79e4681484696b"}}