Abstract
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 W[1]-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 8 k vertices on strongly connected digraphs and O ( k 2 ) vertices on general digraphs. We also give an algorithm for this problem on general digraphs with running time O ( 5.94 2 k â n O ( 1 ) ) , where n is the number of vertices in the input digraph.
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