Tight bounds and a fast FPT algorithm for directed Max-Leaf Spanning Tree

An out-tree T of a directed graph D is a rooted tree subgraph with all arcs directed outwards from the root. An out-branching is a spanning out-tree. By ℓ(D) and ℓ s (D) we denote the maximum number of leaves over all out-trees and out-branchings of D, respectively. We give fixed parameter tractable algorithms for deciding whether ℓ s (D) ≥ k and whether ℓ(D) ≥ k for a digraph D on n vertices, both with time complexity 2 O(klogk) ·n O(1). This proves the problem for out-branchings to be in FPT, and improves on the previous complexity of 2O(klog2 k) ·nO(1)2^{O(k\log^2 k)} \cdot n^{O(1)} for out-trees. To obtain the algorithm for out-branchings, we prove that when all arcs of D are part of at least one out-branching, ℓ s (D) ≥ ℓ(D)/3. The second bound we prove states that for strongly connected digraphs D with minimum in-degree 3, ls(D) ³ Q(Ön)\ell_s(D)\geq \Theta(\sqrt{n}) , where previously ls(D) ³ Q(3Ö{n})\ell_s(D)\geq \Theta(\sqrt[3]{n}) was the best known bound. This bound is tight, and also holds for the larger class of digraphs with minimum in-degree 3 in which every arc is part of at least one out-branching.