{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:MBUM7BMAHUMOR6Z7D45I3NJOYF","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":"11bf2b45e744fedeb4ca72f2f9e7c7ec72d30c95ea9514c8becc6f40a8c96f7f","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2019-07-17T09:48:34Z","title_canon_sha256":"6b58edb782c496f89db4a021bdd0803c41d81567679578ab84fd207a3982e227"},"schema_version":"1.0","source":{"id":"1907.07414","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.07414","created_at":"2026-05-17T23:40:22Z"},{"alias_kind":"arxiv_version","alias_value":"1907.07414v1","created_at":"2026-05-17T23:40:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.07414","created_at":"2026-05-17T23:40:22Z"},{"alias_kind":"pith_short_12","alias_value":"MBUM7BMAHUMO","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_16","alias_value":"MBUM7BMAHUMOR6Z7","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_8","alias_value":"MBUM7BMA","created_at":"2026-05-18T12:33:21Z"}],"graph_snapshots":[{"event_id":"sha256:932d288c580e4f2b39e9bf65c435eb125f53b733b097658f25ec80c630d264d9","target":"graph","created_at":"2026-05-17T23:40:22Z","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":"In this paper, we introduce the notion of the containment graph of a family of sets and containment classes of graphs and posets. Let $Z$ be a family of nonempty sets. We call a (simple, finite) graph G = (V, E) a $Z$-containment graph provided one can assign to each vertex $v_i \\in V $ a set $S_i \\in Z$ such that $v_i v_j \\in E$ if and only if $S_i \\subset S_j$ or $S_j \\subset S_i$ . Similarly, we call a (strict) partially ordered set $P = (V, <)$ a $Z$-containment poset if to each $v_i \\in V $ we can assign a set $S_i \\in Z$ such that $v_i < v_j$ if and only if $S_i \\subset S_j$. Obviously, ","authors_text":"Edward R. Scheinerman, Martin Charles Golumbic","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2019-07-17T09:48:34Z","title":"Containment Graphs, Posets, and Related Classes of Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.07414","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":"sha256:5d25ea8d3b551f7bd5e748526fb918257b68dbef9ca38a788b8c91bf5c7c1d51","target":"record","created_at":"2026-05-17T23:40:22Z","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":"11bf2b45e744fedeb4ca72f2f9e7c7ec72d30c95ea9514c8becc6f40a8c96f7f","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2019-07-17T09:48:34Z","title_canon_sha256":"6b58edb782c496f89db4a021bdd0803c41d81567679578ab84fd207a3982e227"},"schema_version":"1.0","source":{"id":"1907.07414","kind":"arxiv","version":1}},"canonical_sha256":"6068cf85803d18e8fb3f1f3a8db52ec14c719f257655c92b624fde55f44ee2e8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6068cf85803d18e8fb3f1f3a8db52ec14c719f257655c92b624fde55f44ee2e8","first_computed_at":"2026-05-17T23:40:22.319973Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:40:22.319973Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lwYUA7LYjN6gjzKZSgRVhDL3jvyVm0sPeSgXqbQd2mWgCCaqsRIJRefsy6csX5uuKrgs5ZZe8Dx2Brffso9YDA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:40:22.320807Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.07414","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5d25ea8d3b551f7bd5e748526fb918257b68dbef9ca38a788b8c91bf5c7c1d51","sha256:932d288c580e4f2b39e9bf65c435eb125f53b733b097658f25ec80c630d264d9"],"state_sha256":"fa4b826607ada6d1fc9ad937d498750a6fa5d1b9d7ba8ff3b8d8b0724582bc44"}