Spanning directed trees with many leaves
N Alon, F.V. Fomin, G. Gutin, M. Krivelevich, S. Saurabh

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 -vertex digraph with minimum in-degree at least 3 has an out-branching with at least leaves; - if a strongly connected digraph does not contain an out-branching with leaves, then the pathwidth of its underlying graph UG() is . Moreover, if the digraph is acyclic, the pathwidth is at most . The last result implies that it can be decided in time whether a strongly connected digraph on vertices has an out-branching with at least leaves. On acyclic digraphs the running time of our algorithm…
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Taxonomy
TopicsAdvanced Graph Theory Research · Complexity and Algorithms in Graphs · Interconnection Networks and Systems
