{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:J3VSTU24BJVLXQUKIUTE5IDCOH","short_pith_number":"pith:J3VSTU24","canonical_record":{"source":{"id":"1407.6178","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2014-07-23T11:20:24Z","cross_cats_sorted":[],"title_canon_sha256":"30e3fea125c199db02a775e4a700036fce4fb85a36d097f44ccb3d3e50d09af7","abstract_canon_sha256":"d171e0bcc6ed95786861e68d992b84b2c1f80a3fe62a4e734cc17c4e0d9d93bb"},"schema_version":"1.0"},"canonical_sha256":"4eeb29d35c0a6abbc28a45264ea06271de73623512156ee21a69f954af8d9c0a","source":{"kind":"arxiv","id":"1407.6178","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.6178","created_at":"2026-05-18T02:46:58Z"},{"alias_kind":"arxiv_version","alias_value":"1407.6178v1","created_at":"2026-05-18T02:46:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.6178","created_at":"2026-05-18T02:46:58Z"},{"alias_kind":"pith_short_12","alias_value":"J3VSTU24BJVL","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"J3VSTU24BJVLXQUK","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"J3VSTU24","created_at":"2026-05-18T12:28:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:J3VSTU24BJVLXQUKIUTE5IDCOH","target":"record","payload":{"canonical_record":{"source":{"id":"1407.6178","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2014-07-23T11:20:24Z","cross_cats_sorted":[],"title_canon_sha256":"30e3fea125c199db02a775e4a700036fce4fb85a36d097f44ccb3d3e50d09af7","abstract_canon_sha256":"d171e0bcc6ed95786861e68d992b84b2c1f80a3fe62a4e734cc17c4e0d9d93bb"},"schema_version":"1.0"},"canonical_sha256":"4eeb29d35c0a6abbc28a45264ea06271de73623512156ee21a69f954af8d9c0a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:46:58.770371Z","signature_b64":"RjHH1kjIuoF/7C0fj9A/jq6NylZD9OtG5xcxxdidNe7YpfLqN1UQItCf1uop4rlMXbuyWLte/RMFaollmeBCDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4eeb29d35c0a6abbc28a45264ea06271de73623512156ee21a69f954af8d9c0a","last_reissued_at":"2026-05-18T02:46:58.768948Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:46:58.768948Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1407.6178","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":"2026-05-18T02:46:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YnPYjDyhYPoKWlDUKwj/yyK1EZZHAjb9nTyAEYad9m8x7qt0XCwRG4jObKl235io8gdYVIqgNX+4ikyEayi7DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T03:40:22.413102Z"},"content_sha256":"df276cf00d543c45068f1517132279f1e34ee741301d17a526d9a5a57e12603b","schema_version":"1.0","event_id":"sha256:df276cf00d543c45068f1517132279f1e34ee741301d17a526d9a5a57e12603b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:J3VSTU24BJVLXQUKIUTE5IDCOH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Computing the $2$-blocks of directed graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Raed Jaberi","submitted_at":"2014-07-23T11:20:24Z","abstract_excerpt":"Let $G$ be a directed graph. A \\textit{$2$-directed block} in $G$ is a maximal vertex set $C^{2d}\\subseteq V$ with $|C^{2d}|\\geq 2$ such that for each pair of distinct vertices $x,y \\in C^{2d}$, there exist two vertex-disjoint paths from $x$ to $y$ and two vertex-disjoint paths from $y$ to $x$ in $G$. In contrast to the $2$-vertex-connected components of $G$, the subgraphs induced by the $2$-directed blocks may consist of few or no edges. In this paper we present two algorithms for computing the $2$-directed blocks of $G$ in $O(\\min\\lbrace m,(t_{sap}+t_{sb})n\\rbrace n)$ time, where $t_{sap}$ i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6178","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":"2026-05-18T02:46:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3PnIC9ViR1WokNgWeSfkA+yZ6eIUzBM7FLgdz5WcZLTODnkH63o4WQZYHaPbUVU8POfxluClwgpPVp8VSvE1DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T03:40:22.413487Z"},"content_sha256":"f1003ddf759900cbe3b125651c35ba62403a360494fc29cdc02ba6c2de028f68","schema_version":"1.0","event_id":"sha256:f1003ddf759900cbe3b125651c35ba62403a360494fc29cdc02ba6c2de028f68"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/J3VSTU24BJVLXQUKIUTE5IDCOH/bundle.json","state_url":"https://pith.science/pith/J3VSTU24BJVLXQUKIUTE5IDCOH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/J3VSTU24BJVLXQUKIUTE5IDCOH/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":"2026-06-26T03:40:22Z","links":{"resolver":"https://pith.science/pith/J3VSTU24BJVLXQUKIUTE5IDCOH","bundle":"https://pith.science/pith/J3VSTU24BJVLXQUKIUTE5IDCOH/bundle.json","state":"https://pith.science/pith/J3VSTU24BJVLXQUKIUTE5IDCOH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/J3VSTU24BJVLXQUKIUTE5IDCOH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:J3VSTU24BJVLXQUKIUTE5IDCOH","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":"d171e0bcc6ed95786861e68d992b84b2c1f80a3fe62a4e734cc17c4e0d9d93bb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2014-07-23T11:20:24Z","title_canon_sha256":"30e3fea125c199db02a775e4a700036fce4fb85a36d097f44ccb3d3e50d09af7"},"schema_version":"1.0","source":{"id":"1407.6178","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.6178","created_at":"2026-05-18T02:46:58Z"},{"alias_kind":"arxiv_version","alias_value":"1407.6178v1","created_at":"2026-05-18T02:46:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.6178","created_at":"2026-05-18T02:46:58Z"},{"alias_kind":"pith_short_12","alias_value":"J3VSTU24BJVL","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"J3VSTU24BJVLXQUK","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"J3VSTU24","created_at":"2026-05-18T12:28:33Z"}],"graph_snapshots":[{"event_id":"sha256:f1003ddf759900cbe3b125651c35ba62403a360494fc29cdc02ba6c2de028f68","target":"graph","created_at":"2026-05-18T02:46:58Z","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":"Let $G$ be a directed graph. A \\textit{$2$-directed block} in $G$ is a maximal vertex set $C^{2d}\\subseteq V$ with $|C^{2d}|\\geq 2$ such that for each pair of distinct vertices $x,y \\in C^{2d}$, there exist two vertex-disjoint paths from $x$ to $y$ and two vertex-disjoint paths from $y$ to $x$ in $G$. In contrast to the $2$-vertex-connected components of $G$, the subgraphs induced by the $2$-directed blocks may consist of few or no edges. In this paper we present two algorithms for computing the $2$-directed blocks of $G$ in $O(\\min\\lbrace m,(t_{sap}+t_{sb})n\\rbrace n)$ time, where $t_{sap}$ i","authors_text":"Raed Jaberi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2014-07-23T11:20:24Z","title":"Computing the $2$-blocks of directed graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6178","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:df276cf00d543c45068f1517132279f1e34ee741301d17a526d9a5a57e12603b","target":"record","created_at":"2026-05-18T02:46:58Z","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":"d171e0bcc6ed95786861e68d992b84b2c1f80a3fe62a4e734cc17c4e0d9d93bb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2014-07-23T11:20:24Z","title_canon_sha256":"30e3fea125c199db02a775e4a700036fce4fb85a36d097f44ccb3d3e50d09af7"},"schema_version":"1.0","source":{"id":"1407.6178","kind":"arxiv","version":1}},"canonical_sha256":"4eeb29d35c0a6abbc28a45264ea06271de73623512156ee21a69f954af8d9c0a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4eeb29d35c0a6abbc28a45264ea06271de73623512156ee21a69f954af8d9c0a","first_computed_at":"2026-05-18T02:46:58.768948Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:46:58.768948Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RjHH1kjIuoF/7C0fj9A/jq6NylZD9OtG5xcxxdidNe7YpfLqN1UQItCf1uop4rlMXbuyWLte/RMFaollmeBCDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:46:58.770371Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.6178","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:df276cf00d543c45068f1517132279f1e34ee741301d17a526d9a5a57e12603b","sha256:f1003ddf759900cbe3b125651c35ba62403a360494fc29cdc02ba6c2de028f68"],"state_sha256":"6098ba57e3d25e71818c73b6a664cb3bae402ef009a4f16a8a8b559bc66797b6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5FSpY8djIVE7TgYsn6ocuHONmg1qnfPI1YnNo1nZRn8xQliu1ChSE4tNcFDCRYaXOR/E5VJqo1BmkDsy4D8TAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T03:40:22.416075Z","bundle_sha256":"745b7eb2366e87a7823d6cf9398279bf8725ad8c41a889f30dae349ee5b9ebcb"}}