{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:ZFOUCBMUWJZYOAAPG2M2VSMLM4","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":"196237255c4cfc8e68597df8ba45c1472a4fa66e70267347496f790a218e8951","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2016-07-05T15:22:34Z","title_canon_sha256":"ec165e954e39edcdead28e4c87355633df7680961d7b55ab4610b34d2d5ec7b9"},"schema_version":"1.0","source":{"id":"1607.01294","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.01294","created_at":"2026-05-18T01:03:32Z"},{"alias_kind":"arxiv_version","alias_value":"1607.01294v2","created_at":"2026-05-18T01:03:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.01294","created_at":"2026-05-18T01:03:32Z"},{"alias_kind":"pith_short_12","alias_value":"ZFOUCBMUWJZY","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"ZFOUCBMUWJZYOAAP","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"ZFOUCBMU","created_at":"2026-05-18T12:30:53Z"}],"graph_snapshots":[{"event_id":"sha256:89b42fc81583a28bc735ff09cc05469264bf478167f72a05b0bbef88f24bdec8","target":"graph","created_at":"2026-05-18T01:03:32Z","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":"Given a plane forest $F = (V, E)$ of $|V| = n$ points, we find the minimum set $S \\subseteq E$ of edges such that the edge-constrained minimum spanning tree over the set $V$ of vertices and the set $S$ of constraints contains $F$. We present an $O(n \\log n )$-time algorithm that solves this problem. We generalize this to other proximity graphs in the constraint setting, such as the relative neighbourhood graph, Gabriel graph, $\\beta$-skeleton and Delaunay triangulation. We present an algorithm that identifies the minimum set $S\\subseteq E$ of edges of a given plane graph $I=(V,E)$ such that $I","authors_text":"Alina Shaikhet, Jean-Lou De Carufel, Michiel Smid, Prosenjit Bose","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2016-07-05T15:22:34Z","title":"Essential Constraints of Edge-Constrained Proximity Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.01294","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":"sha256:4c475daa65bc6ace831fc9592a801604c7cf045e3fca2801d8f3d8776a1d0312","target":"record","created_at":"2026-05-18T01:03:32Z","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":"196237255c4cfc8e68597df8ba45c1472a4fa66e70267347496f790a218e8951","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2016-07-05T15:22:34Z","title_canon_sha256":"ec165e954e39edcdead28e4c87355633df7680961d7b55ab4610b34d2d5ec7b9"},"schema_version":"1.0","source":{"id":"1607.01294","kind":"arxiv","version":2}},"canonical_sha256":"c95d410594b27387000f3699aac98b6718e76f32eeb687cbc15f8d0a7039f2d3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c95d410594b27387000f3699aac98b6718e76f32eeb687cbc15f8d0a7039f2d3","first_computed_at":"2026-05-18T01:03:32.385958Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:03:32.385958Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9AF4YAWwY8279CN2tBCbV0+0gfNeTaBgD+mjODWxsAIzxfOpfcJ70wogA9ABZkvJJ7fgIXGyajyPe9Smo1GeBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:03:32.386507Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.01294","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4c475daa65bc6ace831fc9592a801604c7cf045e3fca2801d8f3d8776a1d0312","sha256:89b42fc81583a28bc735ff09cc05469264bf478167f72a05b0bbef88f24bdec8"],"state_sha256":"043f3236371db1f5cb8ac408d4a8653439600a3f2cc98181802cf0f8e5835a45"}