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Also, we give a necessary and sufficient condition on $G$ so that $J(G)^{(k)}$ is a componentwise linear ideal for some (equivalently, for all) $k \\geq 2$ when $G$ is a graph such that $G \\setminus N_G[A]$ has a simplicial vertex for any independen"},"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":"2107.00739","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2021-07-01T20:34:16Z","cross_cats_sorted":[],"title_canon_sha256":"07a9ee1b0b49b229aa99c8fed3945dbea9fce4730263b95f87cec60fcef7a688","abstract_canon_sha256":"c613d7f05906c41100a1f24b1e726a22843128a1b63f59866171a10fa3b1dc9e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:57:22.702536Z","signature_b64":"fHM7KE0DIQNKD7bkH8puR/aQpuxnFdUTyBP/ub4lSFXhZG3s3cBnXmjMV7Pbj9OXfgXYQQk9wscHVz4quHI6AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"93dbe915baf07d150c197948a5467397339617747d218062892b44c3f71c7675","last_reissued_at":"2026-07-05T04:57:22.702116Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:57:22.702116Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Componentwise Linearity of Powers of Cover Ideals","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Joseph W. 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