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k-Distinct In- and Out-Branchings in Digraphs Gutin, Gregory Reidl, Felix Wahlström, Magnus Digraphs Branchings Decompositions FPT algorithms An out-branching and an in-branching of a digraph D are called k-distinct if each of them has k arcs absent in the other. Bang-Jensen, Saurabh and Simonsen (2016) proved that the problem of deciding whether a strongly connected digraph D has k-distinct out-branching and in-branching is fixed-parameter tractable (FPT) when parameterized by k. They asked whether the problem remains FPT when extended to arbitrary digraphs. Bang-Jensen and Yeo (2008) asked whether the same problem is FPT when the out-branching and in-branching have the same root. By linking the two problems with the problem of whether a digraph has an out-branching with at least k leaves (a leaf is a vertex of out-degree zero), we first solve the problem of Bang-Jensen and Yeo (2008). We then develop a new digraph decomposition called the rooted cut decomposition and using it we prove that the problem of Bang-Jensen et al. (2016) is FPT for all digraphs. We believe that the rooted cut decomposition will be useful for obtaining other results on digraphs. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Gregory Gutin and Felix Reidl and Magnus Wahlström 2017 Is Part Of LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2017.58 urn:nbn:de:0030-drops-73788 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.58 eng https://creativecommons.org/licenses/by/3.0/legalcode