$k$-Distinct In- and Out-Branchings in Digraphs

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 \emph{rooted cut decomposition} will be useful for obtaining other results on digraphs.