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It is known that if $\\mathcal{H}$ has bounded vertex-cover number (equivalently, the size of the maximum matching in $\\mathcal{H}$ is bounded), then #Sub$(\\mathcal{H})$ is polynomial-time solvable. We complement this result with a corresponding lower bound: if $\\mathcal{H}$ is any recursively enumerable class of graphs with unbounded vertex-cover number, then #Sub$(\\mathcal{H})$ is #W[1]-ha"},"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":"1407.2929","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2014-07-10T19:59:19Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"f259bb1b06178f8937c61d478e7efa8fbac4d2b6c813827d6b0de285e6fff086","abstract_canon_sha256":"8744b21fdd6a6cebf8cb79735eaf93157a0f4d126568f140a1838280413d068d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:47:53.381217Z","signature_b64":"S/ZcvgEXWpmqEkLRfaiqHgI8G/+q6u9vfDDkrvqGZZ5LHZb+WUArPsroI93/B+3yTdVV13dQb8RGsapjcpChAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5d48a70d5d7bfcf4a3ddc53eb309a762609b646defee37d446f5069feaec7b9b","last_reissued_at":"2026-05-18T02:47:53.380647Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:47:53.380647Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Complexity of counting subgraphs: only the boundedness of the vertex-cover number counts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"cs.CC","authors_text":"D\\'aniel Marx, Radu Curticapean","submitted_at":"2014-07-10T19:59:19Z","abstract_excerpt":"For a class $\\mathcal{H}$ of graphs, #Sub$(\\mathcal{H})$ is the counting problem that, given a graph $H\\in \\mathcal{H}$ and an arbitrary graph $G$, asks for the number of subgraphs of $G$ isomorphic to $H$. 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