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A closed graph is a simple graph satisfying the following property: for all edges $\\{i, j\\}$ and $\\{k, \\ell\\}$ with $i < j$ and $k < \\ell$ one has $\\{j, \\ell\\}\\in E(G)$ if $i = k$, and $\\{i, k\\}\\in E(G)$ if $j = \\ell$. We state some criteria for the closedness of a graph $G$ that do not depend necessarily from the labelling of its vertex set. Consequently, if $S = K[x_1,..., x_n, y_1,..., y_n]$ is a polynomial ring in $2n$ variables with coefficients in a field "},"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":"1007.4383","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-07-26T07:24:23Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"6782dd8fb8ad0db7aec6beb202ab2a31567dd5cc7aedf921e8c39822bd3ad576","abstract_canon_sha256":"d8f924e817d2f19c847cf93169a25cbb64283e556253b287f6dbe33cdce24fa2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:12:18.612267Z","signature_b64":"2LxsAmqwlnOBYZXD2VORHv4qrj5UsyxiPxu2vukEQR/DxU4ABBrosRu7YnBr0Ut5RN893OFRBj3Bp+UedmzCAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"42d04168f251c61095aa8bdbf8c93c539949df285ff706ecbcb4c650e63312fd","last_reissued_at":"2026-05-18T04:12:18.611727Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:12:18.611727Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Koszulness of binomial edge ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Giancarlo Rinaldo, Marilena Crupi","submitted_at":"2010-07-26T07:24:23Z","abstract_excerpt":"Let $G$ be a simple graph on the vertex set $V(G) = [n] = \\{1,...,n\\}$ and edge ideal $E(G)$. 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