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This relaxation is crucial in various real-world applications such as link prediction, community detection, and social network analysis. Although the problems of enumerating maximal\n                    <jats:italic toggle=\"yes\">k<\/jats:italic>\n                    -defective cliques and searching a maximum\n                    <jats:italic toggle=\"yes\">k<\/jats:italic>\n                    -defective clique have been extensively studied, existing algorithms suffer from limitations such as the combinatorial explosion of small partial solutions and sub-optimal search spaces. To address these limitations, we propose a novel clique-first branch-and-bound framework that first generates cliques and then adds missing edges. Furthermore, we introduce a new pivoting technique that achieves a search space size of\n                    <jats:italic toggle=\"yes\">O<\/jats:italic>\n                    (3\n                    <jats:sup>n\/3<\/jats:sup>\n                    \u2022 n\n                    <jats:sup>k<\/jats:sup>\n                    ), where\n                    <jats:italic toggle=\"yes\">n<\/jats:italic>\n                    is the number of vertices in the input graph. We prove that the worst-case number of maximal\n                    <jats:italic toggle=\"yes\">k<\/jats:italic>\n                    -defective cliques is \u03a9(3\n                    <jats:sup>n\/3<\/jats:sup>\n                    \u2022 n\n                    <jats:sup>k<\/jats:sup>\n                    ) when\n                    <jats:italic toggle=\"yes\">k<\/jats:italic>\n                    is a constant, establishing that our algorithm's search space is\n                    <jats:italic toggle=\"yes\">worst-case optimal.<\/jats:italic>\n                    Leveraging the diameter-two property of defective cliques, we further reduce the search space size to\n                    <jats:italic toggle=\"yes\">O<\/jats:italic>\n                    (n \u2022 3\n                    <jats:sup>\u03b4\/3<\/jats:sup>\n                    \u2022 (\u03b4 \u0394)\n                    <jats:sup>k<\/jats:sup>\n                    ), where \u03b4 is the degeneracy and \u0394 is the maximum degree of the input graph. We also propose an efficient framework for maximum\n                    <jats:italic toggle=\"yes\">k<\/jats:italic>\n                    -defective clique search based on our branch-and-bound, together with practical techniques to reduce the search space. Experiments on real-world benchmark datasets with more than 1 million edges demonstrate that each of our proposed algorithms for maximal\n                    <jats:italic toggle=\"yes\">k<\/jats:italic>\n                    -defective clique enumeration and maximum\n                    <jats:italic toggle=\"yes\">k<\/jats:italic>\n                    -defective clique search outperforms the respective state-of-the-art algorithms by up to four orders of magnitude in terms of processing time.\n                  <\/jats:p>","DOI":"10.1145\/3769787","type":"journal-article","created":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T04:32:13Z","timestamp":1764995533000},"page":"1-28","source":"Crossref","is-referenced-by-count":1,"title":["Efficient Defective Clique Enumeration and Search with Worst-Case Optimal Search Space"],"prefix":"10.1145","volume":"3","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-6658-3789","authenticated-orcid":false,"given":"Jihoon","family":"Jang","sequence":"first","affiliation":[{"name":"Seoul National University, Seoul, Republic of Korea"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-2561-3335","authenticated-orcid":false,"given":"Yehyun","family":"Nam","sequence":"additional","affiliation":[{"name":"Seoul National University, Seoul, Republic of Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5225-0907","authenticated-orcid":false,"given":"Kunsoo","family":"Park","sequence":"additional","affiliation":[{"name":"Seoul National University, Seoul, Republic of Korea"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-4019-312X","authenticated-orcid":false,"given":"Hyunjoon","family":"Kim","sequence":"additional","affiliation":[{"name":"Hanyang University, Seoul, Republic of Korea"}]}],"member":"320","published-online":{"date-parts":[[2025,12,5]]},"reference":[{"key":"e_1_2_1_1_1","first-page":"598","article-title":"Massive quasi-clique detection","author":"Abello James","year":"2002","unstructured":"James Abello, Mauricio GC Resende, and Sandra Sudarsky. 2002. 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