{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:7SUCCMA4EDCF3YNKND5UCTJGMV","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":"ac2df6eee7261c55417cee01c3f5f7fcdad0e92c8bed3d6d32c4f691f7ae5195","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-03-21T04:20:20Z","title_canon_sha256":"fbc2ac40ec7fca2519ecd0ab6b0743eeeabb4a67ebb60a28ed924de6f939da76"},"schema_version":"1.0","source":{"id":"1003.3968","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.3968","created_at":"2026-05-18T02:24:36Z"},{"alias_kind":"arxiv_version","alias_value":"1003.3968v3","created_at":"2026-05-18T02:24:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.3968","created_at":"2026-05-18T02:24:36Z"},{"alias_kind":"pith_short_12","alias_value":"7SUCCMA4EDCF","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"7SUCCMA4EDCF3YNK","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"7SUCCMA4","created_at":"2026-05-18T12:26:05Z"}],"graph_snapshots":[{"event_id":"sha256:5dcdc1d81eb20375899c5e22babaf1ea3dbaf2103eab5f4385d599cb87e3dd99","target":"graph","created_at":"2026-05-18T02:24:36Z","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 discuss how to find the well-covered dimension of a graph that is the Cartesian product of paths, cycles, complete graphs, and other simple graphs. Also, a bound for the well-covered dimension of $K_n\\times G$ is found, provided that $G$ has a largest greedy independent decomposition of length $c<n$.\n  Formulae to find the well-covered dimension of graphs obtained by vertex blowups on a known graph, and to the lexicographic product of two known graphs are also given.","authors_text":"Isaac Birnbaum, Katherine Urabe, Megan Kuneli, Oscar Vega, Robyn McDonald","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-03-21T04:20:20Z","title":"The Well-Covered Dimension of Products of Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.3968","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:fb8fc7c1edadccc20dcbf84b7e59c3a2e9d857d8e6f9cf9be8aa118247baf7e8","target":"record","created_at":"2026-05-18T02:24:36Z","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":"ac2df6eee7261c55417cee01c3f5f7fcdad0e92c8bed3d6d32c4f691f7ae5195","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-03-21T04:20:20Z","title_canon_sha256":"fbc2ac40ec7fca2519ecd0ab6b0743eeeabb4a67ebb60a28ed924de6f939da76"},"schema_version":"1.0","source":{"id":"1003.3968","kind":"arxiv","version":3}},"canonical_sha256":"fca821301c20c45de1aa68fb414d2665701d41575b74e62079a6f9f7af6690a4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fca821301c20c45de1aa68fb414d2665701d41575b74e62079a6f9f7af6690a4","first_computed_at":"2026-05-18T02:24:36.434072Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:24:36.434072Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kxvweGJMzJn3T66pEzBjqmZ+YfHwXeAbpsBNTZkIuPKF8GcTbj+UCUa0yICm7lJCo8CqRGJEqgMMJDnaTHBdAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:24:36.434698Z","signed_message":"canonical_sha256_bytes"},"source_id":"1003.3968","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb8fc7c1edadccc20dcbf84b7e59c3a2e9d857d8e6f9cf9be8aa118247baf7e8","sha256:5dcdc1d81eb20375899c5e22babaf1ea3dbaf2103eab5f4385d599cb87e3dd99"],"state_sha256":"0218070b51e76ee7ffd5cd27ee7cc25d5f3cc50fb3716926e89f34b70eba3aa3"}