Edge-disjoint in- and out-branchings in tournaments and related path problems
Journal of Combinatorial Theory Series B
Journal of Graph Theory
Digraphs: Theory, Algorithms and Applications
Digraphs: Theory, Algorithms and Applications
Disjoint directed and undirected paths and cycles in digraphs
Theoretical Computer Science
The np-completeness of the hamiltonian cycle problem in planar diagraphs with degree bound two
Information Processing Letters
Arc-disjoint spanning sub(di)graphs in digraphs
Theoretical Computer Science
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An out-branching (in-branching) B"s^+(B"s^-) in a digraph D is a connected spanning subdigraph of D in which every vertex xs has precisely one arc entering (leaving) it and s has no arcs entering (leaving) it. We settle the complexity of the following two problems: *Given a 2-regular digraph D, decide whether it contains two arc-disjoint branchings B"u^+, B"v^-. *Given a 2-regular digraph D, decide whether it contains an out-branching B"u^+ such that D remains connected after removing the arcs of B"u^+. Both problems are NP-complete for general digraphs (Bang-Jensen (1991) [1], Bang-Jensen and Yeo (2012) [7]). We prove that the first problem remains NP-complete for 2-regular digraphs, whereas the second problem turns out to be polynomial when we do not prescribe the root in advance. The complexity when the root is prescribed in advance is still open. We also prove that, for 2-regular digraphs, the second problem is in fact equivalent to deciding whether D contains two arc-disjoint out-branchings.