Spanning Directed Trees with Many Leaves

The 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 (1) 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; (2) if a strongly connected digraph $D$ does not contain an out-branching with $k$ leaves, then the pathwidth of its underlying graph $\mathrm{UG}(D)$ is $O(k\log k)$, and if the digraph is acyclic with a single vertex of in-degree zero, then the pathwidth is at most $4k$. The last result implies that it can be decided in time $2^{O(k\log^2k)}\cdot n^{O(1)}$ 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 $2^{O(k\log k)}\cdot n^{O(1)}$.