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arXiv · 0803.0701

Spanning directed trees with many leaves

N Alon·F. V. Fomin·G. Gutin·M. Krivelevich·S. Saurabh
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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 $2^{O(k\log k)}\cdot n^{O(1)}$.

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N Alon, F. V. Fomin, G. Gutin, M. Krivelevich, S. Saurabh. 2008-03-05. Spanning directed trees with many leaves. https://arxiv.org/abs/0803.0701

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