Parameterized Algorithms for Directed Maximum Leaf Problems

TL;DR

This paper proves fixed parameter tractability for finding rooted subtrees with many leaves in directed graphs, solving open problems and generalizing spanning tree leaf results.

Contribution

It establishes fixed parameter algorithms for directed maximum leaf problems and generalizes spanning tree leaf results to broader digraph classes.

Findings

Rooted subtree with at least k leaves is FPT in digraphs.

Existence of a rooted spanning tree with many leaves in certain digraphs.

Generalization of undirected spanning tree leaf results to directed graphs.

Abstract

We prove that finding a rooted subtree with at least $k$ leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family $\cal L$ that includes all strong and acyclic digraphs. This settles completely an open question of Fellows and solves another one for digraphs in $\cal L$. Our algorithms are based on the following combinatorial result which can be viewed as a generalization of many results for a `spanning tree with many leaves' in the undirected case, and which is interesting on its own: If a digraph $D\in \cal L$ of order $n$ with minimum in-degree at least 3 contains a rooted spanning tree, then $D$ contains one with at least $(n/2)^{1/5}-1$ leaves.