{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:A226A6NZLBB7A7ASDOM6RA5GDV","short_pith_number":"pith:A226A6NZ","canonical_record":{"source":{"id":"2409.12199","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DM","submitted_at":"2024-09-09T07:38:15Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"9e0c9dfdbe1a6e8f903e687d3dbe6c917c5ac067099eb0ce955e2ea730ca52dc","abstract_canon_sha256":"c17d81206502a54b706432427d87d00933050d072b608ad275f2f5145d1ecdbd"},"schema_version":"1.0"},"canonical_sha256":"06b5e079b95843f07c121b99e883a61d7ddd33e6f58b138735743987b5e6fa64","source":{"kind":"arxiv","id":"2409.12199","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2409.12199","created_at":"2026-05-22T01:03:40Z"},{"alias_kind":"arxiv_version","alias_value":"2409.12199v1","created_at":"2026-05-22T01:03:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.12199","created_at":"2026-05-22T01:03:40Z"},{"alias_kind":"pith_short_12","alias_value":"A226A6NZLBB7","created_at":"2026-05-22T01:03:40Z"},{"alias_kind":"pith_short_16","alias_value":"A226A6NZLBB7A7AS","created_at":"2026-05-22T01:03:40Z"},{"alias_kind":"pith_short_8","alias_value":"A226A6NZ","created_at":"2026-05-22T01:03:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:A226A6NZLBB7A7ASDOM6RA5GDV","target":"record","payload":{"canonical_record":{"source":{"id":"2409.12199","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DM","submitted_at":"2024-09-09T07:38:15Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"9e0c9dfdbe1a6e8f903e687d3dbe6c917c5ac067099eb0ce955e2ea730ca52dc","abstract_canon_sha256":"c17d81206502a54b706432427d87d00933050d072b608ad275f2f5145d1ecdbd"},"schema_version":"1.0"},"canonical_sha256":"06b5e079b95843f07c121b99e883a61d7ddd33e6f58b138735743987b5e6fa64","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:03:40.429131Z","signature_b64":"RN7ByiMnEjw9tjcq0jeLifoH6R1uRMFyjwXY6hzwohvdDIQdgSgMPjMDbESNeNau+RrK/ftk1pxUyLH6Is1eBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"06b5e079b95843f07c121b99e883a61d7ddd33e6f58b138735743987b5e6fa64","last_reissued_at":"2026-05-22T01:03:40.428241Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:03:40.428241Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2409.12199","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":"2026-05-22T01:03:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DM8y7IM/fdjW6i6R5H6wSK69qoMElkUJMhU42in+TgkYohdOycl7g1iVI2wjjnAFkBI8u94X7dRASp1DWskJCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T11:17:56.521742Z"},"content_sha256":"e8e1e1fd13fd0325dab9445d66722cf0dcf8e405608585a835b19c6ded55298c","schema_version":"1.0","event_id":"sha256:e8e1e1fd13fd0325dab9445d66722cf0dcf8e405608585a835b19c6ded55298c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:A226A6NZLBB7A7ASDOM6RA5GDV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fault Tolerant Metric Dimensions of Leafless Cacti Graphs with Application in Supply Chain Management","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Faisal Yousafzai, Ghulam Haidar, Murad Ul Islam Khan, Qaisar Khan, Rakea Fatima, Tauseef Asif","submitted_at":"2024-09-09T07:38:15Z","abstract_excerpt":"A resolving set for a simple graph $G$ is a subset of vertex set of $G$ such that it distinguishes all vertices of $G$ using the shortest distance from this subset. This subset is a metric basis if it is the smallest set with this property. A resolving set is a fault tolerant resolving set if the removal of any vertex from the subset still leaves it a resolving set. The smallest set satisfying this property is the fault tolerant metric basis, and the cardinality of this set is termed as fault tolerant metric dimension of $G$, denoted by $\\beta'(G)$. In this article, we determine the fault tole"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.12199","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/2409.12199/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":"2026-05-22T01:03:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EldHuPchoMy/B8yhL6Ge7ic7OS5PRRCLrDAfn5nXJczCrFa9iUnVvFzMktqO2kHJij7PBfGwFqw9bW0+sI3ECQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T11:17:56.522533Z"},"content_sha256":"2695f8965a5473fcce3fc46a7a201544a1cd50071e4439b5d366ec69326a3b04","schema_version":"1.0","event_id":"sha256:2695f8965a5473fcce3fc46a7a201544a1cd50071e4439b5d366ec69326a3b04"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/A226A6NZLBB7A7ASDOM6RA5GDV/bundle.json","state_url":"https://pith.science/pith/A226A6NZLBB7A7ASDOM6RA5GDV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/A226A6NZLBB7A7ASDOM6RA5GDV/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":"2026-06-11T11:17:56Z","links":{"resolver":"https://pith.science/pith/A226A6NZLBB7A7ASDOM6RA5GDV","bundle":"https://pith.science/pith/A226A6NZLBB7A7ASDOM6RA5GDV/bundle.json","state":"https://pith.science/pith/A226A6NZLBB7A7ASDOM6RA5GDV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/A226A6NZLBB7A7ASDOM6RA5GDV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:A226A6NZLBB7A7ASDOM6RA5GDV","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":"c17d81206502a54b706432427d87d00933050d072b608ad275f2f5145d1ecdbd","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DM","submitted_at":"2024-09-09T07:38:15Z","title_canon_sha256":"9e0c9dfdbe1a6e8f903e687d3dbe6c917c5ac067099eb0ce955e2ea730ca52dc"},"schema_version":"1.0","source":{"id":"2409.12199","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2409.12199","created_at":"2026-05-22T01:03:40Z"},{"alias_kind":"arxiv_version","alias_value":"2409.12199v1","created_at":"2026-05-22T01:03:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.12199","created_at":"2026-05-22T01:03:40Z"},{"alias_kind":"pith_short_12","alias_value":"A226A6NZLBB7","created_at":"2026-05-22T01:03:40Z"},{"alias_kind":"pith_short_16","alias_value":"A226A6NZLBB7A7AS","created_at":"2026-05-22T01:03:40Z"},{"alias_kind":"pith_short_8","alias_value":"A226A6NZ","created_at":"2026-05-22T01:03:40Z"}],"graph_snapshots":[{"event_id":"sha256:2695f8965a5473fcce3fc46a7a201544a1cd50071e4439b5d366ec69326a3b04","target":"graph","created_at":"2026-05-22T01:03:40Z","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/2409.12199/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A resolving set for a simple graph $G$ is a subset of vertex set of $G$ such that it distinguishes all vertices of $G$ using the shortest distance from this subset. This subset is a metric basis if it is the smallest set with this property. A resolving set is a fault tolerant resolving set if the removal of any vertex from the subset still leaves it a resolving set. The smallest set satisfying this property is the fault tolerant metric basis, and the cardinality of this set is termed as fault tolerant metric dimension of $G$, denoted by $\\beta'(G)$. In this article, we determine the fault tole","authors_text":"Faisal Yousafzai, Ghulam Haidar, Murad Ul Islam Khan, Qaisar Khan, Rakea Fatima, Tauseef Asif","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DM","submitted_at":"2024-09-09T07:38:15Z","title":"Fault Tolerant Metric Dimensions of Leafless Cacti Graphs with Application in Supply Chain Management"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.12199","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:e8e1e1fd13fd0325dab9445d66722cf0dcf8e405608585a835b19c6ded55298c","target":"record","created_at":"2026-05-22T01:03:40Z","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":"c17d81206502a54b706432427d87d00933050d072b608ad275f2f5145d1ecdbd","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DM","submitted_at":"2024-09-09T07:38:15Z","title_canon_sha256":"9e0c9dfdbe1a6e8f903e687d3dbe6c917c5ac067099eb0ce955e2ea730ca52dc"},"schema_version":"1.0","source":{"id":"2409.12199","kind":"arxiv","version":1}},"canonical_sha256":"06b5e079b95843f07c121b99e883a61d7ddd33e6f58b138735743987b5e6fa64","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"06b5e079b95843f07c121b99e883a61d7ddd33e6f58b138735743987b5e6fa64","first_computed_at":"2026-05-22T01:03:40.428241Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-22T01:03:40.428241Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RN7ByiMnEjw9tjcq0jeLifoH6R1uRMFyjwXY6hzwohvdDIQdgSgMPjMDbESNeNau+RrK/ftk1pxUyLH6Is1eBg==","signature_status":"signed_v1","signed_at":"2026-05-22T01:03:40.429131Z","signed_message":"canonical_sha256_bytes"},"source_id":"2409.12199","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e8e1e1fd13fd0325dab9445d66722cf0dcf8e405608585a835b19c6ded55298c","sha256:2695f8965a5473fcce3fc46a7a201544a1cd50071e4439b5d366ec69326a3b04"],"state_sha256":"379d273e09b07541b90c2343233b509a669c0aa79ab320cffb172cf6710a24c6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mtIMD7qial8SmQfM+mLBsITSO/7MIwmAEqD1+qUsRBo/Q81uANIkFkgub9SpkjOJaGuoRjcq0sf58nrf5pUkAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T11:17:56.526207Z","bundle_sha256":"99dcbcfe9f2f217604f6bab8beb388bebc2400d765c1fa878b0898bbf34f8d7d"}}