{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:SQQM2ZCD2ARDTIREZY5PREBCXU","short_pith_number":"pith:SQQM2ZCD","canonical_record":{"source":{"id":"1806.00302","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-01T12:07:05Z","cross_cats_sorted":[],"title_canon_sha256":"39c241d76118f6de15345a6018ad0396b7f51b09c5c376d01fd362c0a7654f3d","abstract_canon_sha256":"781e13301b5cdd119d23d2ede3dfd5c09fbf4f80e5ba3ed02feec65e1b274b17"},"schema_version":"1.0"},"canonical_sha256":"9420cd6443d02239a224ce3af89022bd3c998bcb936dfca1de12585a91be0840","source":{"kind":"arxiv","id":"1806.00302","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.00302","created_at":"2026-05-18T00:14:24Z"},{"alias_kind":"arxiv_version","alias_value":"1806.00302v1","created_at":"2026-05-18T00:14:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.00302","created_at":"2026-05-18T00:14:24Z"},{"alias_kind":"pith_short_12","alias_value":"SQQM2ZCD2ARD","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_16","alias_value":"SQQM2ZCD2ARDTIRE","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_8","alias_value":"SQQM2ZCD","created_at":"2026-05-18T12:32:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:SQQM2ZCD2ARDTIREZY5PREBCXU","target":"record","payload":{"canonical_record":{"source":{"id":"1806.00302","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-01T12:07:05Z","cross_cats_sorted":[],"title_canon_sha256":"39c241d76118f6de15345a6018ad0396b7f51b09c5c376d01fd362c0a7654f3d","abstract_canon_sha256":"781e13301b5cdd119d23d2ede3dfd5c09fbf4f80e5ba3ed02feec65e1b274b17"},"schema_version":"1.0"},"canonical_sha256":"9420cd6443d02239a224ce3af89022bd3c998bcb936dfca1de12585a91be0840","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:24.105226Z","signature_b64":"flNQeVqgeozH855/rcmrlWT4hnBVGjgFCwcT/siC5oAzsETMz6yqpncs4/VuF+TIdlUwNdeDyAvAci3R58l7Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9420cd6443d02239a224ce3af89022bd3c998bcb936dfca1de12585a91be0840","last_reissued_at":"2026-05-18T00:14:24.104591Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:24.104591Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1806.00302","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":"2026-05-18T00:14:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9Phbnmxf6ipSBSK90fiCYvDPJDq5Aj76X7i78PzITMYow1BVkI1dJwQ2lGvMu8cQ5oR0kNWIwCH3bXlORurSDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T06:02:31.846220Z"},"content_sha256":"535ca43029984e21a9a30cdd1fcaa68acdac136f924923864e0b6a3e319c2f08","schema_version":"1.0","event_id":"sha256:535ca43029984e21a9a30cdd1fcaa68acdac136f924923864e0b6a3e319c2f08"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:SQQM2ZCD2ARDTIREZY5PREBCXU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Strong geodetic problem on complete multipartite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Matja\\v{z} Konvalinka, Vesna Ir\\v{s}i\\v{c}","submitted_at":"2018-06-01T12:07:05Z","abstract_excerpt":"The strong geodetic problem is to find the smallest number of vertices such that by fixing one shortest path between each pair, all vertices of the graph are covered. In this paper we study the strong geodetic problem on complete bipartite graphs; in particular, we discuss its asymptotic behavior. Some results for complete multipartite graphs are also derived. Finally, we prove that the strong geodetic problem restricted to (general) bipartite graphs is NP-complete."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.00302","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":"2026-05-18T00:14:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T+8gCMthGgOjzyswVMiA0S+Xn4fUKOWw093SkKqfn73OoKzkunPm7jnUMdBbvcHlOVi+pnkMac0gAPoMfKDaAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T06:02:31.846949Z"},"content_sha256":"b2ac1725eec17f379a7070e48ff2b1d3d0852154249ea309fd7f3c0ce8e5183d","schema_version":"1.0","event_id":"sha256:b2ac1725eec17f379a7070e48ff2b1d3d0852154249ea309fd7f3c0ce8e5183d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SQQM2ZCD2ARDTIREZY5PREBCXU/bundle.json","state_url":"https://pith.science/pith/SQQM2ZCD2ARDTIREZY5PREBCXU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SQQM2ZCD2ARDTIREZY5PREBCXU/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":"2026-06-11T06:02:31Z","links":{"resolver":"https://pith.science/pith/SQQM2ZCD2ARDTIREZY5PREBCXU","bundle":"https://pith.science/pith/SQQM2ZCD2ARDTIREZY5PREBCXU/bundle.json","state":"https://pith.science/pith/SQQM2ZCD2ARDTIREZY5PREBCXU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SQQM2ZCD2ARDTIREZY5PREBCXU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:SQQM2ZCD2ARDTIREZY5PREBCXU","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":"781e13301b5cdd119d23d2ede3dfd5c09fbf4f80e5ba3ed02feec65e1b274b17","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-01T12:07:05Z","title_canon_sha256":"39c241d76118f6de15345a6018ad0396b7f51b09c5c376d01fd362c0a7654f3d"},"schema_version":"1.0","source":{"id":"1806.00302","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.00302","created_at":"2026-05-18T00:14:24Z"},{"alias_kind":"arxiv_version","alias_value":"1806.00302v1","created_at":"2026-05-18T00:14:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.00302","created_at":"2026-05-18T00:14:24Z"},{"alias_kind":"pith_short_12","alias_value":"SQQM2ZCD2ARD","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_16","alias_value":"SQQM2ZCD2ARDTIRE","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_8","alias_value":"SQQM2ZCD","created_at":"2026-05-18T12:32:53Z"}],"graph_snapshots":[{"event_id":"sha256:b2ac1725eec17f379a7070e48ff2b1d3d0852154249ea309fd7f3c0ce8e5183d","target":"graph","created_at":"2026-05-18T00:14:24Z","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":"The strong geodetic problem is to find the smallest number of vertices such that by fixing one shortest path between each pair, all vertices of the graph are covered. In this paper we study the strong geodetic problem on complete bipartite graphs; in particular, we discuss its asymptotic behavior. Some results for complete multipartite graphs are also derived. Finally, we prove that the strong geodetic problem restricted to (general) bipartite graphs is NP-complete.","authors_text":"Matja\\v{z} Konvalinka, Vesna Ir\\v{s}i\\v{c}","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-01T12:07:05Z","title":"Strong geodetic problem on complete multipartite graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.00302","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":"sha256:535ca43029984e21a9a30cdd1fcaa68acdac136f924923864e0b6a3e319c2f08","target":"record","created_at":"2026-05-18T00:14:24Z","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":"781e13301b5cdd119d23d2ede3dfd5c09fbf4f80e5ba3ed02feec65e1b274b17","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-01T12:07:05Z","title_canon_sha256":"39c241d76118f6de15345a6018ad0396b7f51b09c5c376d01fd362c0a7654f3d"},"schema_version":"1.0","source":{"id":"1806.00302","kind":"arxiv","version":1}},"canonical_sha256":"9420cd6443d02239a224ce3af89022bd3c998bcb936dfca1de12585a91be0840","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9420cd6443d02239a224ce3af89022bd3c998bcb936dfca1de12585a91be0840","first_computed_at":"2026-05-18T00:14:24.104591Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:24.104591Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"flNQeVqgeozH855/rcmrlWT4hnBVGjgFCwcT/siC5oAzsETMz6yqpncs4/VuF+TIdlUwNdeDyAvAci3R58l7Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:24.105226Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.00302","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:535ca43029984e21a9a30cdd1fcaa68acdac136f924923864e0b6a3e319c2f08","sha256:b2ac1725eec17f379a7070e48ff2b1d3d0852154249ea309fd7f3c0ce8e5183d"],"state_sha256":"a4bed6f46dff5aa44df152f33b8890e9aef52dd50db3f6971ce0a53757d1cb6d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f8dSx66Fa+P9vsj/mq28KxYpVK0H0kRHcHskwt6a4wLp4zVSiTbECti+AE0KHPoMFj+N+UBUO5EXVWYvWO8LDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T06:02:31.852298Z","bundle_sha256":"a13aa86911ed3baec232db75cb9d2c2911f4203d821f2be09f05329ff1e5eb1c"}}