Algorithm for finding k-vertex out-trees and its application to k-internal out-branching problem

An out-tree T is an oriented tree with only one vertex of in-degree zero. A vertex x of T is internal if its out-degree is positive. We design randomized and deterministic algorithms for deciding whether an input digraph contains a given out-tree with k vertices. The algorithms are of running time O*(5.704k) and O*(6.14k), respectively. We apply the deterministic algorithm to obtain a deterministic algorithm of runtime O*(ck), where c is a constant, for deciding whether an input digraph contains a spanning out-tree with at least k internal vertices. This answers in affirmative a question of Gutin, Razgon and Kim (Proc. AAIM'08) [9]