Acyclic orientations of mixed graphs

2026-09-26

AI Summary

This study addresses the acyclic orientation completion problem for mixed graphs, which involves determining whether undirected edges can be oriented such that the resulting graph remains acyclic while satisfying specific structural properties. Drawing upon combinatorial optimization and computational complexity theory, it systematically analyzes algorithmic boundaries under various constraints. The primary contributions are threefold. First, it proves that orientations containing a specified single-source out-arborescence are decidable in polynomial time, whereas those involving both in- and out-arborescences, as well as deciding path existence between two vertices, are NP-complete. Second, it reveals a fundamental complexity gap between single-source out-arborescence orientations and bipolar orientations. Finally, it formulates the non-separating out-arborescence orientation as an open problem, thereby identifying new directions for future research in this domain.

Abstract

A mixed graph $M=(V,E\cup A)$ is acyclic if its directed part $(V,A)$ is an acyclic digraph. In this note we study the so-called orientation completion problem for the class of acyclic mixed graphs. That is, given an acyclic mixed graph $M$ and a property ${\cal P}$; can we orient the edges of $M$ so that the resulting digraph is acyclic and has property ${\cal P}$. We prove that one can decide in polynomial time whether $M$ can be completed to an acyclic digraph with an out-branching from a prescibed vertex $s$, while it is NP-complete to decide whether $M$ has an acyclic orientation which has both an out-branching and an in-branching (a bipolar orientation). We show that it is NP-complete to decide whether $M$ can be oriented so that it contains a directed path between two prescribed vertices. Finally we describe a polynomial algorithm for deciding whether an acyclic digraph $D$ has an out-branching $B^+_s$ such that the digraph $D-A(B^+_s)$ is connected (in the underlying sense). Based on this we pose as an open problem the complexity of deciding whether the edges of an acyclic mixed graph can be oriented so that the result is an acyclic digraph with a non-separating out-branching.

Problem

• mixed graphs
• acyclic orientations
• orientation completion
• branching
• computational complexity

Innovation

• mixed graphs
• acyclic orientations
• orientation completion
• bipolar orientation
• out-branching