Tight Bounds and Faster Algorithms for Directed Max-Leaf Problems

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 ls(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 ls(D) ≥ k and whether l(D) ≥ k for a digraph D on n vertices, both with time complexity 2 log k) · n. This improves on previous algorithms with complexity 2 3 log k) · n and 2 log 2 k) · n, respectively. To obtain the complexity bound in the case of out-branchings, we prove that when all arcs of D are part of at least one out-branching, ls(D) ≥ l(D)/3. The second bound we prove in this paper states that for strongly connected digraphs D with minimum in-degree 3, ls(D) ≥ Θ( √ n), where previously ls(D) ≥ Θ( 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.