{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:2TZZ2GFTLFV46QG34GZXGX2QHC","short_pith_number":"pith:2TZZ2GFT","canonical_record":{"source":{"id":"1608.04812","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2016-08-16T23:51:56Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"eab158329149dd46928e64c48610f327f5e58dab89db4512ee2b1f698b6d60a6","abstract_canon_sha256":"9a143b0cd98c74bc0abd13b1435b968254a2c8922cc1118e8604f2bd1d0034ff"},"schema_version":"1.0"},"canonical_sha256":"d4f39d18b3596bcf40dbe1b3735f5038be6bf390fe430733c69d3acb8feab533","source":{"kind":"arxiv","id":"1608.04812","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.04812","created_at":"2026-05-18T01:03:17Z"},{"alias_kind":"arxiv_version","alias_value":"1608.04812v2","created_at":"2026-05-18T01:03:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.04812","created_at":"2026-05-18T01:03:17Z"},{"alias_kind":"pith_short_12","alias_value":"2TZZ2GFTLFV4","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"2TZZ2GFTLFV46QG3","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"2TZZ2GFT","created_at":"2026-05-18T12:29:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:2TZZ2GFTLFV46QG34GZXGX2QHC","target":"record","payload":{"canonical_record":{"source":{"id":"1608.04812","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2016-08-16T23:51:56Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"eab158329149dd46928e64c48610f327f5e58dab89db4512ee2b1f698b6d60a6","abstract_canon_sha256":"9a143b0cd98c74bc0abd13b1435b968254a2c8922cc1118e8604f2bd1d0034ff"},"schema_version":"1.0"},"canonical_sha256":"d4f39d18b3596bcf40dbe1b3735f5038be6bf390fe430733c69d3acb8feab533","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:17.548489Z","signature_b64":"LbHmbHTqtHlzVKkFY4uWbtRXUsauDcwULOifcuFJYfBFz7WmL6xDHezTcnRBoK6bfmmKOKb2oJQlPC0RrxftCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d4f39d18b3596bcf40dbe1b3735f5038be6bf390fe430733c69d3acb8feab533","last_reissued_at":"2026-05-18T01:03:17.547970Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:17.547970Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1608.04812","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:03:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wZS4k72BtnZObKuAQGsCAsbIfmquJAbtd9fivmx1AJsiUhUBMNbO4Cq27VLnfXiGA56110Qdhac0iQWN6yrkDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T06:32:12.037065Z"},"content_sha256":"03088062b21536b93d09828d1012ee0e89680586ea73aea645d0234263b9b046","schema_version":"1.0","event_id":"sha256:03088062b21536b93d09828d1012ee0e89680586ea73aea645d0234263b9b046"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:2TZZ2GFTLFV46QG34GZXGX2QHC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Monotone Paths in Geometric Triangulations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CG","authors_text":"Adrian Dumitrescu, Csaba D. T\\'oth, Ritankar Mandal","submitted_at":"2016-08-16T23:51:56Z","abstract_excerpt":"(I) We prove that the (maximum) number of monotone paths in a geometric triangulation of $n$ points in the plane is $O(1.7864^n)$. This improves an earlier upper bound of $O(1.8393^n)$; the current best lower bound is $\\Omega(1.7003^n)$.\n  (II) Given a planar geometric graph $G$ with $n$ vertices, we show that the number of monotone paths in $G$ can be computed in $O(n^2)$ time."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.04812","kind":"arxiv","version":2},"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":"2026-05-18T01:03:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O12yElG3k/Tknf0VjgVv/8PBf4dtBc7oCkQ8+cpE85BIESdzAn169swuN0ZpPrHFEhmGtn3fO4jnxjrvjlFuCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T06:32:12.037834Z"},"content_sha256":"ae537a887bc60ade40f1404b922f577d7655d10454680ba05a0a42007fdc3b83","schema_version":"1.0","event_id":"sha256:ae537a887bc60ade40f1404b922f577d7655d10454680ba05a0a42007fdc3b83"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2TZZ2GFTLFV46QG34GZXGX2QHC/bundle.json","state_url":"https://pith.science/pith/2TZZ2GFTLFV46QG34GZXGX2QHC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2TZZ2GFTLFV46QG34GZXGX2QHC/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":"2026-05-30T06:32:12Z","links":{"resolver":"https://pith.science/pith/2TZZ2GFTLFV46QG34GZXGX2QHC","bundle":"https://pith.science/pith/2TZZ2GFTLFV46QG34GZXGX2QHC/bundle.json","state":"https://pith.science/pith/2TZZ2GFTLFV46QG34GZXGX2QHC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2TZZ2GFTLFV46QG34GZXGX2QHC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:2TZZ2GFTLFV46QG34GZXGX2QHC","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":"9a143b0cd98c74bc0abd13b1435b968254a2c8922cc1118e8604f2bd1d0034ff","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2016-08-16T23:51:56Z","title_canon_sha256":"eab158329149dd46928e64c48610f327f5e58dab89db4512ee2b1f698b6d60a6"},"schema_version":"1.0","source":{"id":"1608.04812","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.04812","created_at":"2026-05-18T01:03:17Z"},{"alias_kind":"arxiv_version","alias_value":"1608.04812v2","created_at":"2026-05-18T01:03:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.04812","created_at":"2026-05-18T01:03:17Z"},{"alias_kind":"pith_short_12","alias_value":"2TZZ2GFTLFV4","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"2TZZ2GFTLFV46QG3","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"2TZZ2GFT","created_at":"2026-05-18T12:29:55Z"}],"graph_snapshots":[{"event_id":"sha256:ae537a887bc60ade40f1404b922f577d7655d10454680ba05a0a42007fdc3b83","target":"graph","created_at":"2026-05-18T01:03:17Z","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":"(I) We prove that the (maximum) number of monotone paths in a geometric triangulation of $n$ points in the plane is $O(1.7864^n)$. This improves an earlier upper bound of $O(1.8393^n)$; the current best lower bound is $\\Omega(1.7003^n)$.\n  (II) Given a planar geometric graph $G$ with $n$ vertices, we show that the number of monotone paths in $G$ can be computed in $O(n^2)$ time.","authors_text":"Adrian Dumitrescu, Csaba D. T\\'oth, Ritankar Mandal","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2016-08-16T23:51:56Z","title":"Monotone Paths in Geometric Triangulations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.04812","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":"sha256:03088062b21536b93d09828d1012ee0e89680586ea73aea645d0234263b9b046","target":"record","created_at":"2026-05-18T01:03:17Z","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":"9a143b0cd98c74bc0abd13b1435b968254a2c8922cc1118e8604f2bd1d0034ff","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2016-08-16T23:51:56Z","title_canon_sha256":"eab158329149dd46928e64c48610f327f5e58dab89db4512ee2b1f698b6d60a6"},"schema_version":"1.0","source":{"id":"1608.04812","kind":"arxiv","version":2}},"canonical_sha256":"d4f39d18b3596bcf40dbe1b3735f5038be6bf390fe430733c69d3acb8feab533","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d4f39d18b3596bcf40dbe1b3735f5038be6bf390fe430733c69d3acb8feab533","first_computed_at":"2026-05-18T01:03:17.547970Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:03:17.547970Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LbHmbHTqtHlzVKkFY4uWbtRXUsauDcwULOifcuFJYfBFz7WmL6xDHezTcnRBoK6bfmmKOKb2oJQlPC0RrxftCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:03:17.548489Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.04812","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:03088062b21536b93d09828d1012ee0e89680586ea73aea645d0234263b9b046","sha256:ae537a887bc60ade40f1404b922f577d7655d10454680ba05a0a42007fdc3b83"],"state_sha256":"846c9dbce4c491c67720a412cf256861f51f62747e5697e51591540ba8e43f39"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QAtFXpkxtiYtzqV5tCOktt8uzpMIZZPt62IAcfsT/ra8uXc3wKBQ7bDwov/tTE/yURcFdC056Byk9X3ARdhAAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T06:32:12.039943Z","bundle_sha256":"ed185b19f3a0b778bf27bd93b7358880e195b61b7e486534f3422554fca04f4a"}}