{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:SQ64K77NR7VT7KW7M7FJHUAC7W","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":"171e58d1b8ebaac698234eb56e62916056bf410fd7bbe92a473823d3114d450c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-29T11:16:38Z","title_canon_sha256":"9c32f3702db0e33f4dbefa4ba9c36a5fcd686cb317efaf6b81eecb2aa89f3174"},"schema_version":"1.0","source":{"id":"1602.08907","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.08907","created_at":"2026-05-18T01:19:35Z"},{"alias_kind":"arxiv_version","alias_value":"1602.08907v3","created_at":"2026-05-18T01:19:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.08907","created_at":"2026-05-18T01:19:35Z"},{"alias_kind":"pith_short_12","alias_value":"SQ64K77NR7VT","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"SQ64K77NR7VT7KW7","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"SQ64K77N","created_at":"2026-05-18T12:30:44Z"}],"graph_snapshots":[{"event_id":"sha256:ec1866bc45ebc4ef3f771a779871ab7f3cf51a76675beaf5665f0153002afcfd","target":"graph","created_at":"2026-05-18T01:19:35Z","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":"A partition P of the vertex set of a connected graph G is a locating partition of G if every vertex is uniquely determined by its vector of distances to the elements of P. The partition dimension of G is the minimum cardinality of a locating partition of G. A pair of vertices u,v of a graph G are called twins if they have exactly the same set of neighbors other than u and v. A twin class is a maximal set of pairwise twin vertices. The twin number of a graph G is the maximum cardinality of a twin class of G.\n  In this paper we undertake the study of the partition dimension of a graph by also co","authors_text":"Carmen Hernando, Ignacio M Pelayo, Merce Mora","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-29T11:16:38Z","title":"On the Partition Dimension and the Twin Number of a Graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.08907","kind":"arxiv","version":3},"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":"sha256:79a6a69eebf28c883965047fc8b1b44efa052fbfb9afc2aa0923cb2cf5ee0c87","target":"record","created_at":"2026-05-18T01:19:35Z","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":"171e58d1b8ebaac698234eb56e62916056bf410fd7bbe92a473823d3114d450c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-29T11:16:38Z","title_canon_sha256":"9c32f3702db0e33f4dbefa4ba9c36a5fcd686cb317efaf6b81eecb2aa89f3174"},"schema_version":"1.0","source":{"id":"1602.08907","kind":"arxiv","version":3}},"canonical_sha256":"943dc57fed8feb3faadf67ca93d002fdb39f195636a294b43600e52afe8e35d8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"943dc57fed8feb3faadf67ca93d002fdb39f195636a294b43600e52afe8e35d8","first_computed_at":"2026-05-18T01:19:35.136670Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:35.136670Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LUUxTZGTdfxcjg4vQNKXqiHvpOI2pJHF09pN6tOA83ZWIA3Dce06EXJJXo8ES3qZBwFx7CY6IBQ0GIjHVoowDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:35.137096Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.08907","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:79a6a69eebf28c883965047fc8b1b44efa052fbfb9afc2aa0923cb2cf5ee0c87","sha256:ec1866bc45ebc4ef3f771a779871ab7f3cf51a76675beaf5665f0153002afcfd"],"state_sha256":"9ba768b4403574116b1e4f1ac4752184c685ed4e81f0a58a32ecb7700646de9b"}