{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:YMQ4P35EPL5HCK6HB4GK56MZ6D","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":"cac99449fbf0b6df300138964c0b2442e10e80f6eb5ae8afcbba5552c9dd3272","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-17T22:14:10Z","title_canon_sha256":"cf1d7a1d8dc3b3edbb426718b2bd65797b524664444dd6bcd06f6bef9b96fed8"},"schema_version":"1.0","source":{"id":"1501.04236","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.04236","created_at":"2026-05-18T00:45:59Z"},{"alias_kind":"arxiv_version","alias_value":"1501.04236v1","created_at":"2026-05-18T00:45:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.04236","created_at":"2026-05-18T00:45:59Z"},{"alias_kind":"pith_short_12","alias_value":"YMQ4P35EPL5H","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"YMQ4P35EPL5HCK6H","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"YMQ4P35E","created_at":"2026-05-18T12:29:50Z"}],"graph_snapshots":[{"event_id":"sha256:7c7cfb562423cf5f05840c70c790ab6d8283ca8f0e6c722277f8ffa90ba34817","target":"graph","created_at":"2026-05-18T00:45:59Z","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 define three new pebbling parameters of a connected graph $G$, the $r$-, $g$-, and $u$-critical pebbling numbers. Together with the pebbling number, the optimal pebbling number, the number of vertices $n$ and the diameter $d$ of the graph, this yields 7 graph parameters. We determine the relationships between these parameters. We investigate properties of the $r$-critical pebbling number, and distinguish between greedy graphs, thrifty graphs, and graphs for which the $r$-critical pebbling number is $2^d$.","authors_text":"Courtney R. Gibbons, Erick J. Paul, Joshua D. Laison","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-17T22:14:10Z","title":"Critical Pebbling Numbers of Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.04236","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:c998ae1827229feab540040ab8376bc9d5dcb72a75cd4b1073d321dd7931a006","target":"record","created_at":"2026-05-18T00:45:59Z","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":"cac99449fbf0b6df300138964c0b2442e10e80f6eb5ae8afcbba5552c9dd3272","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-17T22:14:10Z","title_canon_sha256":"cf1d7a1d8dc3b3edbb426718b2bd65797b524664444dd6bcd06f6bef9b96fed8"},"schema_version":"1.0","source":{"id":"1501.04236","kind":"arxiv","version":1}},"canonical_sha256":"c321c7efa47afa712bc70f0caef999f0ebda97f74944dc4a0dbec6cd643e60e5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c321c7efa47afa712bc70f0caef999f0ebda97f74944dc4a0dbec6cd643e60e5","first_computed_at":"2026-05-18T00:45:59.014374Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:45:59.014374Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"p7wUU3xQPjqWPiCCGuCUB/LMGKGeTobCC3OW0AQIa8FDU0vLvJhPVR9XvclOoxiIwT0zP7LE13qeF919kH+4DA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:45:59.014855Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.04236","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c998ae1827229feab540040ab8376bc9d5dcb72a75cd4b1073d321dd7931a006","sha256:7c7cfb562423cf5f05840c70c790ab6d8283ca8f0e6c722277f8ffa90ba34817"],"state_sha256":"40b077407ac6e8b557582fb943688cf01bb58a551ebde0d3612f34f117bc16da"}