Arc-disjoint out-branchings and in-branchings in semicomplete digraphs

TL;DR

This paper classifies when semicomplete digraphs contain arc-disjoint out- and in-branchings with prescribed roots, providing a simple characterization and polynomial algorithm, confirming a conjecture for this class.

Contribution

It offers a complete, simple classification and polynomial-time algorithm for the existence of arc-disjoint out- and in-branchings in semicomplete digraphs with prescribed roots.

Findings

Complete classification of semicomplete digraphs with such branchings

Polynomial algorithm for checking the existence of branchings

Generalization of previous tournament characterization

Abstract

An out-branching $B^+_u$ (in-branching $B^-_u$) in a digraph $D$ is a connected spanning subdigraph of $D$ in which every vertex except the vertex $u$, called the root, has in-degree (out-degree) one. It is well-known that there exists a polynomial algorithm for deciding whether a given digraph has $k$ arc-disjoint out-branchings with prescribed roots ($k$ is part of the input). In sharp contrast to this, it is already NP-complete to decide if a digraph has one out-branching which is arc-disjoint from some in-branching. A digraph is {\bf semicomplete} if it has no pair of non adjacent vertices. A {\bf tournament} is a semicomplete digraph without directed cycles of length 2. In this paper we give a complete classification of semicomplete digraphs which have an out-branching $B^+_u$ which is arc-disjoint from some in-branching $B^-_v$ where $u,v$ are prescribed vertices of $D$. Our characterization, which is surprisingly simple, generalizes a complicated characterization for tournaments from 1991 by the first author and our proof implies the existence of a polynomial algorithm for checking whether a given semicomplete digraph has such a pair of branchings for prescribed vertices $u,v$ and construct a solution if one exists. This confirms a conjecture of Bang-Jensen for the case of semicomplete digraphs.