On the Directed Degree-Preserving Spanning Tree Problem

In this paper we initiate a systematic study of the Reduced Degree Spanning Tree problem, where given a digraph D and a nonnegative integer k, the goal is to construct a spanning out-tree with at most k vertices of reduced out-degree. This problem is a directed analog of the wellstudied Minimum-Vertex Feedback Edge Set problem. We show that this problem is fixed-parameter tractable and admits a problem kernel with at most 8k vertices on strongly connected digraphs and O(k) vertices on general digraphs. We also give an algorithm for this problem on general digraphs with runtime O(5.942). This adds the Reduced Degree Spanning Tree problem to the small list of directed graph problems for which fixed-parameter tractable algorithms are known. Finally, we consider the dual of Reduced Degree Spanning Tree, that is, given a digraph D and a nonnegative integer k, the goal is to construct a spanning out-tree of D with at least k vertices of full out-degree. We show that this problem is W[1]-hard on two important digraph classes: directed acyclic graphs and strongly connected digraphs.