{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:7HE7V6A6MPOY6O4XGWLFP2YZXE","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":"44cb07b774450c357372dcc37bad56794a8ef2355ca9abebc80236a861fce94e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-07-06T04:04:01Z","title_canon_sha256":"fe0a3859bfdfe7168678043ab8644dcbadab9bac40776c777b5606676b253a8b"},"schema_version":"1.0","source":{"id":"1007.0804","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.0804","created_at":"2026-05-18T03:41:02Z"},{"alias_kind":"arxiv_version","alias_value":"1007.0804v1","created_at":"2026-05-18T03:41:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.0804","created_at":"2026-05-18T03:41:02Z"},{"alias_kind":"pith_short_12","alias_value":"7HE7V6A6MPOY","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"7HE7V6A6MPOY6O4X","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"7HE7V6A6","created_at":"2026-05-18T12:26:04Z"}],"graph_snapshots":[{"event_id":"sha256:27797e93274d057bb9aa58ebee7f97df9d2f70e8412eaf5733322db60715744d","target":"graph","created_at":"2026-05-18T03:41:02Z","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":"An {\\it overlap representation} of a graph $G$ assigns sets to vertices so that vertices are adjacent if and only if their assigned sets intersect with neither containing the other. The {\\it overlap number} $\\ol(G)$ (introduced by Rosgen) is the minimum size of the union of the sets in such a representation. We prove the following: (1) An optimal overlap representation of a tree can be produced in linear time, and its size is the number of vertices in the largest subtree in which the neighbor of any leaf has degree 2. (2) If $\\delta(G)\\ge 2$ and $G\\ne K_3$, then $\\ol(G)\\le |E(G)|-1$, with equa","authors_text":"Christopher Stocker, Daniel W. Cranston, Douglas B. West, Jennifer Vandenbussche, Kevin Milans, Nitish Korula, Timothy D. LeSaulnier","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-07-06T04:04:01Z","title":"Overlap Number of Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.0804","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":"sha256:94237091cb94b9a46987a18a183af2a772832433cc2373a18b2cd5e731f3e694","target":"record","created_at":"2026-05-18T03:41:02Z","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":"44cb07b774450c357372dcc37bad56794a8ef2355ca9abebc80236a861fce94e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-07-06T04:04:01Z","title_canon_sha256":"fe0a3859bfdfe7168678043ab8644dcbadab9bac40776c777b5606676b253a8b"},"schema_version":"1.0","source":{"id":"1007.0804","kind":"arxiv","version":1}},"canonical_sha256":"f9c9faf81e63dd8f3b97359657eb19b915d2f8dcbcff4a442c8f50f2d1c41a82","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f9c9faf81e63dd8f3b97359657eb19b915d2f8dcbcff4a442c8f50f2d1c41a82","first_computed_at":"2026-05-18T03:41:02.631949Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:41:02.631949Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"E/u/23+HBjkQcwhMaBlRiuveSdidemA8IVHEneFHZTNk51S6ZXkh/m7Ur+X3wIhKW0sfBqQUryCBQNHn6Pm2BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:41:02.632514Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.0804","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:94237091cb94b9a46987a18a183af2a772832433cc2373a18b2cd5e731f3e694","sha256:27797e93274d057bb9aa58ebee7f97df9d2f70e8412eaf5733322db60715744d"],"state_sha256":"a95fde6e62117d1db7bcd2dd0009fc61701770a7e19a818f28417501122aeac9"}