{"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 $H$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.2929","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}