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 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 2O(k log k) •nO(1). This proves the problem for out-branchings to be in FPT, and improves on the previous complexity of 2 O(k log2 k) • nO(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, ℓs(D) ≥ ℓ(D)/3. The second bound we prove 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. © 2008 Springer-Verlag Berlin Heidelberg.