Abstract
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 improve known parameterized algorithms and combinatorial bounds on the number of leaves in out-branchings. We show that
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every strongly connected digraph D of order n with minimum in-degree at least 3 has an out-branching with at least (n/4)1/3− 1 leaves;
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if a strongly connected digraph D does not contain an out-branching with k leaves, then the pathwidth of its underlying graph is O(klogk);
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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.
All improvements use properties of extremal structures obtained after applying local search and properties of some out-branching decompositions.
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Alon, N., Fomin, F.V., Gutin, G., Krivelevich, M., Saurabh, S. (2007). Better Algorithms and Bounds for Directed Maximum Leaf Problems. In: Arvind, V., Prasad, S. (eds) FSTTCS 2007: Foundations of Software Technology and Theoretical Computer Science. FSTTCS 2007. Lecture Notes in Computer Science, vol 4855. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77050-3_26
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DOI: https://doi.org/10.1007/978-3-540-77050-3_26
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