Communication: On complexity of Minimum Leaf Out-Branching problem
Discrete Applied Mathematics
FPT algorithms and kernels for the Directedk- Leaf problem
Journal of Computer and System Sciences
Approximation algorithms for the maximum leaf spanning tree problem on acyclic digraphs
WAOA'11 Proceedings of the 9th international conference on Approximation and Online Algorithms
On the directed Full Degree Spanning Tree problem
Discrete Optimization
A 3/2-Approximation Algorithm for Finding Spanning Trees with Many Leaves in Cubic Graphs
SIAM Journal on Discrete Mathematics
Kernel(s) for problems with no kernel: On out-trees with many leaves
ACM Transactions on Algorithms (TALG)
Beyond bidimensionality: Parameterized subexponential algorithms on directed graphs
Information and Computation
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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 $2^{O(k\log k)}\cdot n^{O(1)}$.