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Listing all maximal cliques in sparse graphs in near-optimal time Eppstein, David Löffler, Maarten Strash, Darren Clique backtracking degeneracy worst-case optimality The degeneracy of an $n$-vertex graph $G$ is the smallest number $d$ such that every subgraph of $G$ contains a vertex of degree at most $d$. We show that there exists a nearly-optimal fixed-parameter tractable algorithm for enumerating all maximal cliques, parametrized by degeneracy. To achieve this result, we modify the classic Bron--Kerbosch algorithm and show that it runs in time $O(dn3^{d/3})$. We also provide matching upper and lower bounds showing that the largest possible number of maximal cliques in an $n$-vertex graph with degeneracy $d$ (when $d$ is a multiple of 3 and $nge d+3$) is $(n-d)3^{d/3}$. Therefore, our algorithm matches the $Theta(d(n-d)3^{d/3})$ worst-case output size of the problem whenever $n-d=Omega(n)$. Schloss Dagstuhl – Leibniz-Zentrum für Informatik David Eppstein and Maarten Löffler and Darren Strash 2011 Is Part Of Dagstuhl Seminar Proceedings, Volume 10441, Exact Complexity of NP-hard Problems (2011) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/DagSemProc.10441.2 urn:nbn:de:0030-drops-29356 https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10441.2 eng https://creativecommons.org/licenses/by/4.0/legalcode