Spanning directed trees with many leaves

TL;DR

This paper investigates the maximum number of leaves in out-branchings of directed graphs, providing combinatorial bounds and an algorithmic approach for determining the existence of out-branchings with many leaves.

Contribution

It presents new bounds on leaves in out-branchings for strongly connected digraphs and develops fixed-parameter algorithms based on pathwidth.

Findings

Every strongly connected digraph with min in-degree ≥ 3 has an out-branching with at least (n/4)^{1/3}-1 leaves.

Absence of an out-branching with k leaves implies the underlying graph's pathwidth is O(k log k).

Deciding the existence of an out-branching with at least k leaves can be done in fixed-parameter tractable time.

Abstract

The {\sc Directed Maximum Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in out-branchings. We show that - every strongly connected $n$-vertex digraph $D$ with minimum in-degree at least 3 has an out-branching with at least $(n/4)^{1/3}-1$ leaves; - if a strongly connected digraph $D$ does not contain an out-branching with $k$ leaves, then the pathwidth of its underlying graph UG($D$) is $O(k\log k)$. Moreover, if the digraph is acyclic, the pathwidth is at most $4k$. The last result implies that it can be decided in time $2^{O(k\log^2 k)}\cdot n^{O(1)}$ whether a strongly connected digraph on $n$ vertices has an out-branching with at least $k$ leaves. On acyclic digraphs the running time of our algorithm is $2^{O(k\log k)}\cdot n^{O(1)}$.