Tight Bounds and a Fast FPT Algorithm for Directed Max-Leaf Spanning Tree

Abstract

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 $2^{O(k\log^2 k)} \cdot n^{O(1)}$ 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, $\ell_s(D)\geq \Theta(\sqrt{n})$, where previously $\ell_s(D)\geq \Theta(\sqrt[3]{n})$ 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.