Arc-Disjoint Paths and Trees in 2-Regular Digraphs
J{\o}rgen Bang-Jensen, Sven Simonsen

TL;DR
This paper investigates the computational complexity of finding arc-disjoint branchings and spanning trees in 2-regular digraphs, establishing NP-completeness results and polynomial cases, and generalizing to k-regular digraphs.
Contribution
It proves NP-completeness of certain problems in 2-regular digraphs and identifies polynomial cases, extending results to k-regular digraphs with multiple disjoint spanning structures.
Findings
NP-complete for two arc-disjoint branchings in 2-regular digraphs
Polynomial-time solution for finding an out-branching maintaining connectivity without fixed root
Generalization to k-regular digraphs with multiple arc-disjoint spanning trees and out-branchings
Abstract
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…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Graph theory and applications
