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We design a meta-algorithm that allows to solve both problems in time $$2^{O(rw^3)}\\cdot n^{4}$$<\/jats:tex-math>\n \n \n 2<\/mml:mn>\n \n O<\/mml:mi>\n (<\/mml:mo>\n r<\/mml:mi>\n \n w<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n \u00b7<\/mml:mo>\n \n n<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, $$2^{O(q^2\\log (q))}\\cdot n^{4}$$<\/jats:tex-math>\n \n \n 2<\/mml:mn>\n \n O<\/mml:mi>\n (<\/mml:mo>\n \n q<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n log<\/mml:mo>\n \n (<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n \u00b7<\/mml:mo>\n \n n<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and $$n^{O(k^2)}$$<\/jats:tex-math>\n \n n<\/mml:mi>\n \n O<\/mml:mi>\n (<\/mml:mo>\n \n k<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> where rw<\/jats:italic> is the rank-width, q<\/jats:italic> the $${\\mathbb {Q}}$$<\/jats:tex-math>\n Q<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-rank-width, and k<\/jats:italic> the mim-width of a given decomposition. This answers in the affirmative an open question raised by Jaffke et\u00a0al. (Algorithmica 82(1):118\u2013145, 2020) concerning an algorithm for SFVS parameterized by mim-width. By a unified algorithm, this solves both problems in polynomial-time on the following graph classes: Interval<\/jats:sc>, Permutation<\/jats:sc>, and Bi-Interval<\/jats:sc> graphs, Circular Arc<\/jats:sc> and Circular Permutation<\/jats:sc> graphs, Convex<\/jats:sc> graphs, k<\/jats:italic>-Polygon<\/jats:sc>, Dilworth<\/jats:sc>-k<\/jats:italic> and Co-<\/jats:sc>k<\/jats:italic>-Degenerate<\/jats:sc> graphs for fixed k<\/jats:italic>; and also on Leaf Power<\/jats:sc> graphs if a leaf root is given as input, on H<\/jats:italic>-Graphs<\/jats:sc> for fixed H<\/jats:italic> if an H<\/jats:italic>-representation is given as input, and on arbitrary powers of graphs in all the above classes. Prior to our results, only SFVS was known to be tractable restricted only on Interval<\/jats:sc> and Permutation<\/jats:sc> graphs, whereas all other results are new. \n<\/jats:p>","DOI":"10.1007\/s00453-022-00936-w","type":"journal-article","created":{"date-parts":[[2022,1,30]],"date-time":"2022-01-30T12:02:32Z","timestamp":1643544152000},"page":"1385-1417","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Node Multiway Cut and Subset Feedback Vertex Set on Graphs of Bounded Mim-Width"],"prefix":"10.1007","volume":"84","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6270-3663","authenticated-orcid":false,"given":"Benjamin","family":"Bergougnoux","sequence":"first","affiliation":[]},{"given":"Charis","family":"Papadopoulos","sequence":"additional","affiliation":[]},{"given":"Jan Arne","family":"Telle","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,1,30]]},"reference":[{"key":"936_CR1","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1016\/j.tcs.2013.01.011","volume":"511","author":"R Belmonte","year":"2013","unstructured":"Belmonte, R., Vatshelle, M.: Graph classes with structured neighborhoods and algorithmic applications. 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