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Arc-Disjoint Paths and Trees in 2-Regular Digraphs
An out-(in-)branching B_s^+ (B_s^-) rooted at s in a digraph D is a connected
spanning subdigraph of D in which every vertex x != s has precisely one arc
entering (leaving) it and s has no arcs entering (leaving) it. We settle the
complexity of the following two problems:
1) Given a 2-regular digraph , decide if it contains two arc-disjoint
branchings B^+_u, B^-_v.
2) Given a 2-regular digraph D, decide if 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. 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. We
also prove that, for 2-regular digraphs, the latter problem is in fact
equivalent to deciding if contains two arc-disjoint out-branchings. We
generalize this result to k-regular digraphs where we want to find a number of
pairwise arc-disjoint spanning trees and out-branchings such that there are k
in total, again without prescribing any roots.Comment: 9 pages, 7 figure
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