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Algorithms"],"published-print":{"date-parts":[[2023,10,31]]},"abstract":"<jats:p>\n            A graph\n            <jats:italic>G<\/jats:italic>\n            is a\n            <jats:italic>k<\/jats:italic>\n            -leaf power if there exists a tree\n            <jats:italic>T<\/jats:italic>\n            whose leaf set is\n            <jats:italic>V<\/jats:italic>\n            (\n            <jats:italic>G<\/jats:italic>\n            ), and such that\n            <jats:italic>uv<\/jats:italic>\n            \u2208\n            <jats:italic>E<\/jats:italic>\n            (\n            <jats:italic>G<\/jats:italic>\n            ) if and only if the distance between\n            <jats:italic>u<\/jats:italic>\n            and\n            <jats:italic>v<\/jats:italic>\n            in\n            <jats:italic>T<\/jats:italic>\n            is at most\n            <jats:italic>k<\/jats:italic>\n            (and\n            <jats:italic>u<\/jats:italic>\n            \u2260\n            <jats:italic>v<\/jats:italic>\n            ). The graph classes of\n            <jats:italic>k<\/jats:italic>\n            -leaf powers have several applications in computational biology, but recognizing them has remained a challenging algorithmic problem for the past two decades. The best known result is that 6-leaf powers can be recognized in polynomial time. In this article, we present an algorithm that decides whether a graph\n            <jats:italic>G<\/jats:italic>\n            is a\n            <jats:italic>k<\/jats:italic>\n            -leaf power in time\n            <jats:italic>O<\/jats:italic>\n            (\n            <jats:italic>n<\/jats:italic>\n            <jats:sup>\n              <jats:italic>f(k)<\/jats:italic>\n            <\/jats:sup>\n            for some function\n            <jats:italic>f<\/jats:italic>\n            that depends only on\n            <jats:italic>k<\/jats:italic>\n            (but has the growth rate of a power tower function).\n          <\/jats:p>\n          <jats:p>\n            Our techniques are based on the fact that either a\n            <jats:italic>k<\/jats:italic>\n            -leaf power has a corresponding tree of low maximum degree, in which case finding it is easy, or every corresponding tree has large maximum degree. In the latter case, large-degree vertices in the tree imply that\n            <jats:italic>G<\/jats:italic>\n            has redundant substructures which can be pruned from the graph. In addition to solving a long-standing open problem, it is our hope that the structural results presented in this work can lead to further results on\n            <jats:italic>k<\/jats:italic>\n            -leaf powers and related classes.\n          <\/jats:p>","DOI":"10.1145\/3614094","type":"journal-article","created":{"date-parts":[[2023,8,8]],"date-time":"2023-08-08T12:02:22Z","timestamp":1691496142000},"page":"1-35","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":5,"title":["Recognizing\n            <i>k<\/i>\n            -Leaf Powers in Polynomial Time, for Constant\n            <i>k<\/i>"],"prefix":"10.1145","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5305-7372","authenticated-orcid":false,"given":"Manuel","family":"Lafond","sequence":"first","affiliation":[{"name":"Universit\u00e9 de Sherbrooke, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2023,9,26]]},"reference":[{"key":"e_1_3_2_2_2","volume-title":"Proceedings of the International Workshop on Graph-Theoretic Concepts in Computer Science","author":"Bergougnoux Benjamin","year":"2022","unstructured":"Benjamin Bergougnoux, Svein H\u00f8gemo, Martin Vatshelle, and Jan Arne Telle. 2022. 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