{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:VKLVC3AEEFBZV7QU5QYISTUJZI","short_pith_number":"pith:VKLVC3AE","canonical_record":{"source":{"id":"2605.19497","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-19T07:51:05Z","cross_cats_sorted":[],"title_canon_sha256":"7e2bebadccd3932d8cf8ca7517cacab51ea3e9fc46635ccce9dcb1316c2f1e80","abstract_canon_sha256":"a342ae6720a612a710c6b0b2c20507295e88caba89711b4c1a079d77ff44ad8b"},"schema_version":"1.0"},"canonical_sha256":"aa97516c0421439afe14ec30894e89ca02ef06c933ce7c8dbb2c105116160c03","source":{"kind":"arxiv","id":"2605.19497","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.19497","created_at":"2026-05-20T01:05:48Z"},{"alias_kind":"arxiv_version","alias_value":"2605.19497v1","created_at":"2026-05-20T01:05:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.19497","created_at":"2026-05-20T01:05:48Z"},{"alias_kind":"pith_short_12","alias_value":"VKLVC3AEEFBZ","created_at":"2026-05-20T01:05:48Z"},{"alias_kind":"pith_short_16","alias_value":"VKLVC3AEEFBZV7QU","created_at":"2026-05-20T01:05:48Z"},{"alias_kind":"pith_short_8","alias_value":"VKLVC3AE","created_at":"2026-05-20T01:05:48Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:VKLVC3AEEFBZV7QU5QYISTUJZI","target":"record","payload":{"canonical_record":{"source":{"id":"2605.19497","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-19T07:51:05Z","cross_cats_sorted":[],"title_canon_sha256":"7e2bebadccd3932d8cf8ca7517cacab51ea3e9fc46635ccce9dcb1316c2f1e80","abstract_canon_sha256":"a342ae6720a612a710c6b0b2c20507295e88caba89711b4c1a079d77ff44ad8b"},"schema_version":"1.0"},"canonical_sha256":"aa97516c0421439afe14ec30894e89ca02ef06c933ce7c8dbb2c105116160c03","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T01:05:48.686869Z","signature_b64":"iDu8X+x/PknLr3iRMTVqAjXqN9cE7rQoSdHDQ2WktAsuluTZOi2kFEN4JEfGMDLS+IGvx+hXKPkhVhTrhaEiCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa97516c0421439afe14ec30894e89ca02ef06c933ce7c8dbb2c105116160c03","last_reissued_at":"2026-05-20T01:05:48.686129Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T01:05:48.686129Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.19497","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":"2026-05-20T01:05:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gpOFrQHgIFaF51fojKWOxdeSsH+tXEnn+EccwCh9oCD1JjhWy9GtxxpBJ8NczIcByImp86nNyqxMKRHP+C6BAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T11:28:21.581218Z"},"content_sha256":"9b3cc3cd1ef432a5d0dded861868cc3addc27e4f805113ae4e4d0f9c45132c27","schema_version":"1.0","event_id":"sha256:9b3cc3cd1ef432a5d0dded861868cc3addc27e4f805113ae4e4d0f9c45132c27"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:VKLVC3AEEFBZV7QU5QYISTUJZI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the disjunctive domination numbers of the torus grid graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Zhi Qiao","submitted_at":"2026-05-19T07:51:05Z","abstract_excerpt":"Let $\\Gamma=(V,E)$ be a graph. A disjunctive dominating set in $\\Gamma$ is a set $S\\subseteq V$ such that every vertex not in $S$ is adjacent to a vertex of $S$, or has at least two vertices in $S$ at distance $2$ from it.\n  The minimum cardinality of a disjunctive dominating set in $\\Gamma$ is called the disjunctive domination number of $\\Gamma$.\n  In this paper, we give bounds for the disjunctive domination numbers of the torus grid graphs $C_m\\Box C_n$, and determine the disjunctive domination numbers of $C_3\\Box C_n$ and $C_4\\Box C_n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.19497","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.19497/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-05-20T01:05:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kQKub1Xn7tjSSSD/D353lhDxMmowwvB5sCCzuRSwG9LsxnZE6wtlb39KCSNSNl4oJ5kgQ/2G3+4PCxC1p1NfCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T11:28:21.581612Z"},"content_sha256":"618a8b3a4679e032ff11dbf62c2e43cd3af9ccdb5bfbfcca820055ba3007e05b","schema_version":"1.0","event_id":"sha256:618a8b3a4679e032ff11dbf62c2e43cd3af9ccdb5bfbfcca820055ba3007e05b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VKLVC3AEEFBZV7QU5QYISTUJZI/bundle.json","state_url":"https://pith.science/pith/VKLVC3AEEFBZV7QU5QYISTUJZI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VKLVC3AEEFBZV7QU5QYISTUJZI/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":"2026-06-02T11:28:21Z","links":{"resolver":"https://pith.science/pith/VKLVC3AEEFBZV7QU5QYISTUJZI","bundle":"https://pith.science/pith/VKLVC3AEEFBZV7QU5QYISTUJZI/bundle.json","state":"https://pith.science/pith/VKLVC3AEEFBZV7QU5QYISTUJZI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VKLVC3AEEFBZV7QU5QYISTUJZI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:VKLVC3AEEFBZV7QU5QYISTUJZI","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":"a342ae6720a612a710c6b0b2c20507295e88caba89711b4c1a079d77ff44ad8b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-19T07:51:05Z","title_canon_sha256":"7e2bebadccd3932d8cf8ca7517cacab51ea3e9fc46635ccce9dcb1316c2f1e80"},"schema_version":"1.0","source":{"id":"2605.19497","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.19497","created_at":"2026-05-20T01:05:48Z"},{"alias_kind":"arxiv_version","alias_value":"2605.19497v1","created_at":"2026-05-20T01:05:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.19497","created_at":"2026-05-20T01:05:48Z"},{"alias_kind":"pith_short_12","alias_value":"VKLVC3AEEFBZ","created_at":"2026-05-20T01:05:48Z"},{"alias_kind":"pith_short_16","alias_value":"VKLVC3AEEFBZV7QU","created_at":"2026-05-20T01:05:48Z"},{"alias_kind":"pith_short_8","alias_value":"VKLVC3AE","created_at":"2026-05-20T01:05:48Z"}],"graph_snapshots":[{"event_id":"sha256:618a8b3a4679e032ff11dbf62c2e43cd3af9ccdb5bfbfcca820055ba3007e05b","target":"graph","created_at":"2026-05-20T01:05:48Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2605.19497/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $\\Gamma=(V,E)$ be a graph. A disjunctive dominating set in $\\Gamma$ is a set $S\\subseteq V$ such that every vertex not in $S$ is adjacent to a vertex of $S$, or has at least two vertices in $S$ at distance $2$ from it.\n  The minimum cardinality of a disjunctive dominating set in $\\Gamma$ is called the disjunctive domination number of $\\Gamma$.\n  In this paper, we give bounds for the disjunctive domination numbers of the torus grid graphs $C_m\\Box C_n$, and determine the disjunctive domination numbers of $C_3\\Box C_n$ and $C_4\\Box C_n$.","authors_text":"Zhi Qiao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-19T07:51:05Z","title":"On the disjunctive domination numbers of the torus grid graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.19497","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":"sha256:9b3cc3cd1ef432a5d0dded861868cc3addc27e4f805113ae4e4d0f9c45132c27","target":"record","created_at":"2026-05-20T01:05:48Z","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":"a342ae6720a612a710c6b0b2c20507295e88caba89711b4c1a079d77ff44ad8b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-19T07:51:05Z","title_canon_sha256":"7e2bebadccd3932d8cf8ca7517cacab51ea3e9fc46635ccce9dcb1316c2f1e80"},"schema_version":"1.0","source":{"id":"2605.19497","kind":"arxiv","version":1}},"canonical_sha256":"aa97516c0421439afe14ec30894e89ca02ef06c933ce7c8dbb2c105116160c03","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aa97516c0421439afe14ec30894e89ca02ef06c933ce7c8dbb2c105116160c03","first_computed_at":"2026-05-20T01:05:48.686129Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T01:05:48.686129Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iDu8X+x/PknLr3iRMTVqAjXqN9cE7rQoSdHDQ2WktAsuluTZOi2kFEN4JEfGMDLS+IGvx+hXKPkhVhTrhaEiCA==","signature_status":"signed_v1","signed_at":"2026-05-20T01:05:48.686869Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.19497","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9b3cc3cd1ef432a5d0dded861868cc3addc27e4f805113ae4e4d0f9c45132c27","sha256:618a8b3a4679e032ff11dbf62c2e43cd3af9ccdb5bfbfcca820055ba3007e05b"],"state_sha256":"a22537ed7d24fed20c5d97f5c6d32edf78235722545150f6bd657428ad66b3c2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MrTBlJKhavndpVVYfIRe3p1RfR8Qo+kstA2Hik8+wXgcV4ud+h7VTjz0D/mfF7WtdT/SIdlf6qbX9K+xiZVfAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T11:28:21.583595Z","bundle_sha256":"979a627c876d078c50429733d569a72ddc5695cad7b3fb18a2444736d75a1290"}}