Out-branchings with Maximal Number of Leaves or Internal Vertices: Algorithmic Results and Open Problems

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 out-tree is called an out-branching. A vertex x of an out-branching B is called a leaf (an internal vertex) if ().In 2005 M. Fellows asked whether it is fixed parameter tractable (FPT) to decide whether a digraph has an out-tree (out-branching, respectively) with at least k leaves, where k is the parameter. Both problems were settled in the affirmative in 2007 by Alon et al., and Bonsma and Dorn, respectively. Since then asymptotically much more efficient FPT algorithms were designed. It is remarkable that the best among them are even more efficient then the best algorithms designed only for undirected graphs.We discuss briefly these and other results including those on the maximum number of internal vertices as well as open problems. Apart from fixed parameter tractability and kernelization, we consider polynomial and approximation algorithms as well as approximation hardness results.