{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:UUYNZ324B64U6T2DTMJPQFLJQK","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":"e5216f36dfdef8d924c8a48eb4e361ba85c58f4778d111204a979dca2aff3df3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-02-03T17:38:15Z","title_canon_sha256":"7123c5441111cbe9490136143e67dc6cd8d271eefc4e825bffbb19a618435f99"},"schema_version":"1.0","source":{"id":"1602.01385","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.01385","created_at":"2026-05-18T01:21:20Z"},{"alias_kind":"arxiv_version","alias_value":"1602.01385v1","created_at":"2026-05-18T01:21:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.01385","created_at":"2026-05-18T01:21:20Z"},{"alias_kind":"pith_short_12","alias_value":"UUYNZ324B64U","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"UUYNZ324B64U6T2D","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"UUYNZ324","created_at":"2026-05-18T12:30:46Z"}],"graph_snapshots":[{"event_id":"sha256:b81bc68b96e46adf02145d2e081f8f17cebc2c9e9349036d5f3df54b6c0b79f4","target":"graph","created_at":"2026-05-18T01:21:20Z","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":"We study the Minimum Shared Edges problem introduced by Omran et al. [Journal of Combinatorial Optimization, 2015] on planar graphs: Planar MSE asks, given a planar graph G = (V,E), two distinct vertices s,t in V , and two integers p, k, whether there are p s-t paths in G that share at most k edges, where an edges is called shared if it appears in at least two of the p s-t paths. We show that Planar MSE is NP-hard by reduction from Vertex Cover. We make use of a grid-like structure, where the alignment (horizontal/vertical) of the edges in the grid correspond to selection and validation gadget","authors_text":"Manuel Sorge, Till Fluschnik","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-02-03T17:38:15Z","title":"The Minimum Shared Edges Problem on Planar Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.01385","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":"sha256:5e2742ec3413079a5e0bb1c66830c8ec040898075459e95f3109d1ad97783c28","target":"record","created_at":"2026-05-18T01:21:20Z","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":"e5216f36dfdef8d924c8a48eb4e361ba85c58f4778d111204a979dca2aff3df3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-02-03T17:38:15Z","title_canon_sha256":"7123c5441111cbe9490136143e67dc6cd8d271eefc4e825bffbb19a618435f99"},"schema_version":"1.0","source":{"id":"1602.01385","kind":"arxiv","version":1}},"canonical_sha256":"a530dcef5c0fb94f4f439b12f81569829e73190ce5201c8c38b2e9d1acea1e21","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a530dcef5c0fb94f4f439b12f81569829e73190ce5201c8c38b2e9d1acea1e21","first_computed_at":"2026-05-18T01:21:20.318809Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:21:20.318809Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BlEYFbOGPTQMGoImHZVwQamvafOw3OYQrB/6gJRUrfMZoZuZ/pEt0qQ4/70vsmuweQthr9PzWoBRyPXm/olwAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:21:20.319562Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.01385","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5e2742ec3413079a5e0bb1c66830c8ec040898075459e95f3109d1ad97783c28","sha256:b81bc68b96e46adf02145d2e081f8f17cebc2c9e9349036d5f3df54b6c0b79f4"],"state_sha256":"3d1eb63795c45a65adaf262952fa66426485a4264e073fc237f7f7b6ff69c56b"}