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Extending a result by the second author, who verified Aharoni's question in the affirmative for chordal graphs, we prove that the answer to the question is yes for all graphs. Namely, we show that for a graph $G$, the non-cover complex of a graph $G$ is $(|V(G)|-i \\gamma(G)-1)$-collapsible"},"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":"1904.10320","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-23T13:37:29Z","cross_cats_sorted":[],"title_canon_sha256":"9e602e9bf0c0777b4c55fd8b4bdd5e0d86a91cba5d779c51a5dd2fc74295c8ec","abstract_canon_sha256":"57571b4679c9c0cf4705a0a423acc2b2760a674f0372b5ddd14f5d1b3311e632"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:47:54.460546Z","signature_b64":"voto472DcezjaFsxOKjx6mhKug5HrnVOpdYSYM0dTyK+y3wTQ7WnHL4TS01Ott/+AuJa0ziM3KB+2+k1jZaxDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"52149f92a7e006e58ac622ea728cde281af0549fd9de60395f773a535a050010","last_reissued_at":"2026-05-17T23:47:54.460021Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:47:54.460021Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Collapsibility of non-cover complexes of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Boram Park, Ilkyoo Choi, Jinha Kim","submitted_at":"2019-04-23T13:37:29Z","abstract_excerpt":"Given a graph $G$, the non-cover complex of $G$ is the combinatorial Alexander dual of the independence complex of $G$. 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