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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"},"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":"1007.0804","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-07-06T04:04:01Z","cross_cats_sorted":[],"title_canon_sha256":"fe0a3859bfdfe7168678043ab8644dcbadab9bac40776c777b5606676b253a8b","abstract_canon_sha256":"44cb07b774450c357372dcc37bad56794a8ef2355ca9abebc80236a861fce94e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:02.632514Z","signature_b64":"E/u/23+HBjkQcwhMaBlRiuveSdidemA8IVHEneFHZTNk51S6ZXkh/m7Ur+X3wIhKW0sfBqQUryCBQNHn6Pm2BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f9c9faf81e63dd8f3b97359657eb19b915d2f8dcbcff4a442c8f50f2d1c41a82","last_reissued_at":"2026-05-18T03:41:02.631949Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:02.631949Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Overlap Number of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christopher Stocker, Daniel W. 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