{"paper":{"title":"Algorithms parameterized by vertex cover and modular width, through potential maximal cliques","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Fedor V. Fomin, Ioan Todinca, Mathieu Liedloff, Pedro Montealegre","submitted_at":"2014-04-15T12:01:55Z","abstract_excerpt":"In this paper we give upper bounds on the number of minimal separators and potential maximal cliques of graphs w.r.t. two graph parameters, namely vertex cover ($\\operatorname{vc}$) and modular width ($\\operatorname{mw}$). We prove that for any graph, the number of minimal separators is $\\mathcal{O}^*(3^{\\operatorname{vc}})$ and $\\mathcal{O}^*(1.6181^{\\operatorname{mw}})$, and the number of potential maximal cliques is $\\mathcal{O}^*(4^{\\operatorname{vc}})$ and $\\mathcal{O}^*(1.7347^{\\operatorname{mw}})$, and these objects can be listed within the same running times. (The $\\mathcal{O}^*$ notat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.3882","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"}