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The goal is to decide whether there exists a sequence $$\\langle I_1, I_2, \\ldots , I_\\ell \\rangle $$<\/jats:tex-math>\n \n \u27e8<\/mml:mo>\n \n I<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n I<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n \n I<\/mml:mi>\n \u2113<\/mml:mi>\n <\/mml:msub>\n \u27e9<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of independent sets such that for all $$j \\in \\{1,\\ldots , \\ell - 1\\}$$<\/jats:tex-math>\n \n j<\/mml:mi>\n \u2208<\/mml:mo>\n {<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n \u2113<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> the set $$I_j$$<\/jats:tex-math>\n \n I<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is an independent set of size\u00a0k<\/jats:italic>, $$I_1 = I_s$$<\/jats:tex-math>\n \n \n I<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n =<\/mml:mo>\n \n I<\/mml:mi>\n s<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, $$I_\\ell = I_t$$<\/jats:tex-math>\n \n \n I<\/mml:mi>\n \u2113<\/mml:mi>\n <\/mml:msub>\n =<\/mml:mo>\n \n I<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$I_j \\triangle I_{j + 1} = \\{u, v\\} \\in E(G)$$<\/jats:tex-math>\n \n \n I<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n \u25b5<\/mml:mi>\n \n I<\/mml:mi>\n \n j<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n =<\/mml:mo>\n \n {<\/mml:mo>\n u<\/mml:mi>\n ,<\/mml:mo>\n v<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n \u2208<\/mml:mo>\n E<\/mml:mi>\n \n (<\/mml:mo>\n G<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Intuitively, we view each independent set as a collection of tokens placed on the vertices of the graph. Then, the problem asks whether there exists a sequence of independent sets that transforms $$I_s$$<\/jats:tex-math>\n \n I<\/mml:mi>\n s<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> into $$I_t$$<\/jats:tex-math>\n \n I<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> where at each step we are allowed to slide one token from a vertex to a neighboring vertex. In this paper, we focus on the parameterized complexity of Token Sliding<\/jats:sc> parameterized by k<\/jats:italic>. As shown by Bartier et al.\u00a0(Algorithmica 83(9):2914\u20132951, 2021. https:\/\/doi.org\/10.1007\/s00453-021-00848-1<\/jats:ext-link>), the problem is -hard on graphs of girth four or less, and the authors posed the question of whether there exists a constant $$p \\ge 5$$<\/jats:tex-math>\n \n p<\/mml:mi>\n \u2265<\/mml:mo>\n 5<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> such that the problem becomes fixed-parameter tractable on graphs of girth at least p<\/jats:italic>. We answer their question positively and prove that the problem is indeed fixed-parameter tractable on graphs of girth five or more, which establishes a full classification of the tractability of Token Sliding<\/jats:sc> parameterized by the number of tokens based on the girth of the input graph.<\/jats:p>","DOI":"10.1007\/s00453-023-01181-5","type":"journal-article","created":{"date-parts":[[2023,10,28]],"date-time":"2023-10-28T12:01:44Z","timestamp":1698494504000},"page":"638-655","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Token Sliding on Graphs of Girth Five"],"prefix":"10.1007","volume":"86","author":[{"given":"Valentin","family":"Bartier","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nicolas","family":"Bousquet","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jihad","family":"Hanna","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Amer E.","family":"Mouawad","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sebastian","family":"Siebertz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,10,28]]},"reference":[{"key":"1181_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.tcs.2016.05.015","volume":"639","author":"RC Brewster","year":"2016","unstructured":"Brewster, R.C., McGuinness, S., Moore, B., Noel, J.A.: A dichotomy theorem for circular colouring reconfiguration. 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