A subgraph T of a digraph D is an out-branching if T is an oriented spanning tree with only one vertex of in-degree zero (called the root). The vertices of T of out-degree zero are leaves. In the Directed k-Leaf Problem, we are given a digraph D and an integral parameter k, and we are to decide whether D has an out-branching with at least k leaves. Recently, Kneis et al. (2008) obtained an algorithm for the problem of running time 4 k · n O(1). We describe a new algorithm for the problem of running time 3.72 k · n O(1). In Rooted Directed k-Leaf Problem, apart from D and k, we are given a vertex r of D and we are to decide whether D has an out-branching rooted at r with at least k leaves. Very recently, Fernau et al. (2008) found an O(k 3)-size kernel for Rooted Directed k-Leaf. In this paper, we obtain an O(k) kernel for Rooted Directed k-Leaf restricted to acyclic digraphs. 1

Abstract. 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 2) vertices on general digraphs. We also give an algorithm for this problem on general digraphs with runtime O ∗ (5.942 k). 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. 1

A subdigraph T of a digraph D is called an out-tree if T is an oriented tree with just one vertex s of in-degree zero. A spanning outtree is called an out-branching. A vertex x of an out-branching B is called a leaf if d + B (x) = 0. This is mainly a survey paper on out-branchings with minimum and maximum number of leaves. We give short proofs of some well-known theorems. 1