{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:HPBAFLWNB6XPK5B2LKQOJGWHVN","short_pith_number":"pith:HPBAFLWN","schema_version":"1.0","canonical_sha256":"3bc202aecd0faef5743a5aa0e49ac7ab4a0ebc420732409bd6930799f6ce5eff","source":{"kind":"arxiv","id":"1901.09859","version":1},"attestation_state":"computed","paper":{"title":"Graphs with a unique maximum open packing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo\\v{s}tjan Bre\\v{s}ar, Douglas F. Rall, Kirsti Kuenzel","submitted_at":"2019-01-28T18:04:31Z","abstract_excerpt":"A set $S$ of vertices in a graph is an open packing if (open) neighborhoods of any two distinct vertices in $S$ are disjoint. In this paper, we consider the graphs that have a unique maximum open packing. We characterize the trees with this property by using four local operations such that any nontrivial tree with a unique maximum open packing can be obtained by a sequence of these operations starting from $P_2$. We also prove that the decision version of the open packing number is NP-complete even when restricted to graphs of girth at least $6$. Finally, we show that the recognition of the gr"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1901.09859","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-01-28T18:04:31Z","cross_cats_sorted":[],"title_canon_sha256":"5dc4fbd08bffe845459ba449a2ff161800a47c9a63265f5166e1d6e1eb17f8f2","abstract_canon_sha256":"ada0186441361d517aa3e91b31cc8faadcff8021c5e9671a640ac8de2b656a70"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:23.649114Z","signature_b64":"BZE3x1t3Vte/NgD4CjaZFGCwMlWnE5SwbBG+Oe6eX0KP+iqWH3+2MpEkgXXNKn/6nuxEZ+i7OBNCSo+pFZIFAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3bc202aecd0faef5743a5aa0e49ac7ab4a0ebc420732409bd6930799f6ce5eff","last_reissued_at":"2026-05-17T23:55:23.648634Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:23.648634Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Graphs with a unique maximum open packing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo\\v{s}tjan Bre\\v{s}ar, Douglas F. Rall, Kirsti Kuenzel","submitted_at":"2019-01-28T18:04:31Z","abstract_excerpt":"A set $S$ of vertices in a graph is an open packing if (open) neighborhoods of any two distinct vertices in $S$ are disjoint. In this paper, we consider the graphs that have a unique maximum open packing. We characterize the trees with this property by using four local operations such that any nontrivial tree with a unique maximum open packing can be obtained by a sequence of these operations starting from $P_2$. We also prove that the decision version of the open packing number is NP-complete even when restricted to graphs of girth at least $6$. Finally, we show that the recognition of the gr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.09859","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1901.09859","created_at":"2026-05-17T23:55:23.648719+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.09859v1","created_at":"2026-05-17T23:55:23.648719+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.09859","created_at":"2026-05-17T23:55:23.648719+00:00"},{"alias_kind":"pith_short_12","alias_value":"HPBAFLWNB6XP","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_16","alias_value":"HPBAFLWNB6XPK5B2","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_8","alias_value":"HPBAFLWN","created_at":"2026-05-18T12:33:18.533446+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HPBAFLWNB6XPK5B2LKQOJGWHVN","json":"https://pith.science/pith/HPBAFLWNB6XPK5B2LKQOJGWHVN.json","graph_json":"https://pith.science/api/pith-number/HPBAFLWNB6XPK5B2LKQOJGWHVN/graph.json","events_json":"https://pith.science/api/pith-number/HPBAFLWNB6XPK5B2LKQOJGWHVN/events.json","paper":"https://pith.science/paper/HPBAFLWN"},"agent_actions":{"view_html":"https://pith.science/pith/HPBAFLWNB6XPK5B2LKQOJGWHVN","download_json":"https://pith.science/pith/HPBAFLWNB6XPK5B2LKQOJGWHVN.json","view_paper":"https://pith.science/paper/HPBAFLWN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.09859&json=true","fetch_graph":"https://pith.science/api/pith-number/HPBAFLWNB6XPK5B2LKQOJGWHVN/graph.json","fetch_events":"https://pith.science/api/pith-number/HPBAFLWNB6XPK5B2LKQOJGWHVN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HPBAFLWNB6XPK5B2LKQOJGWHVN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HPBAFLWNB6XPK5B2LKQOJGWHVN/action/storage_attestation","attest_author":"https://pith.science/pith/HPBAFLWNB6XPK5B2LKQOJGWHVN/action/author_attestation","sign_citation":"https://pith.science/pith/HPBAFLWNB6XPK5B2LKQOJGWHVN/action/citation_signature","submit_replication":"https://pith.science/pith/HPBAFLWNB6XPK5B2LKQOJGWHVN/action/replication_record"}},"created_at":"2026-05-17T23:55:23.648719+00:00","updated_at":"2026-05-17T23:55:23.648719+00:00"}