Arc-Disjoint Paths and Trees in 2-Regular Digraphs

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 $D$, 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 $D$ 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.