FPT algorithms and kernels for the DIRECTED k-LEAF problem

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 MAX LEAF problem, we wish to find the maximum number of leaves in an out-branching of a given digraph D (or, to report that D has no out-branching). 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 DIRECTED k-LEAF of running time 4(k) . no(O(1)). we describe a new algorithm for DIRECTED k-LEAF of running time 3.72(k) . no(O(1)). This algorithms leads to an O(1.9973(n))-time algorithm for solving DIRECTED MAX LEAF on a digraph of order n. The latter algorithm is the first algorithm of running time O (gamma(n)) for DIRECTED MAX LEAF. where gamma < 2. In the 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. (C) 2009 Elsevier Inc. All rights reserved.