Eun Jung Kim

A subgraph $T$ of a digraph $D$ is an {\em out-branching} if $T$ is an oriented spanning tree with only one vertex of in-degree zero (called the {\em root}). The vertices of $T$ of out-degree zero are {\em leaves}. In the {\sc 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 $4^{k}\cdot n^{O(1)}$. We describe a new algorithm for the problem of running time $3.72^{k}\cdot n^{O(1)}$. In {\sc 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 {\sc Rooted Directed $k$-Leaf}. In this paper, we obtain an $O(k)$ kernel for {\sc Rooted Directed $k$-Leaf} restricted to acyclic digraphs.

Given a digraph $D$, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in $D$ an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. We prove that MinLOB is polynomial-time solvable for acyclic digraphs. In general, MinLOB is NP-hard and we consider three parameterizations of MinLOB. We prove that two of them are NP-complete for every value of the parameter, but the third one is fixed-parameter tractable (FPT). The FPT parametrization is as follows: given a digraph $D$ of order $n$ and a positive integral parameter $k$, check whether $D$ contains an out-branching with at most $n-k$ leaves (and find such an out-branching if it exists). We find a problem kernel of order $O(k^2)$ and construct an algorithm of running time $O(2^{O(k\log k)}+n^6),$ which is an `additive' FPT algorithm. We also consider transformations from two related problems, the minimum path covering and the maximum internal out-tree problems into MinLOB, which imply that some parameterizations of the two problems are FPT as well.