On the directed Full Degree Spanning Tree problem

We study the parameterized complexity of a directed analog of the Full Degree Spanning Tree problem where, given a digraph $D$ and a nonnegative integer $k$, the goal is to construct a spanning out-tree $T$ of $D$ such that at least $k$ vertices in $T$ have the same out-degree as in $D$. We show that this problem is W1]-hard even on the class of directed acyclic graphs. In the dual version, called Reduced Degree Spanning Tree, one is required to construct a spanning out-tree $T$ such that at most $k$ vertices in $T$ have out-degrees that are different from that in $D$. We show that this problem is fixed-parameter tractable and that it admits a problem kernel with at most $8k$ vertices on strongly connected digraphs and $O\left(k^2\right)$ vertices on general digraphs. We also give an algorithm for this problem on general digraphs with running time $O(5.942^k?n^{O\left(1\right)})$, where $n$ is the number of vertices in the input digraph.