发表机构
University of Southern Denmark; Shandong University; University of Johannesburg（南丹麦大学; 山东大学; 约翰内斯堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究无环混合图的定向完成问题，证明若干判定问题的复杂性，并给出一个多项式算法，提出开放问题。
AI 中文摘要
一个混合图 $M=(V,E\cup A)$ 是无环的，如果它的有向部分 $(V,A)$ 是一个无环有向图。在本文中，我们研究所谓的一类无环混合图的定向完成问题。即，给定一个无环混合图 $M$ 和一个性质 ${\cal P}$；我们能否对 $M$ 的边进行定向，使得所得有向图是无环的并且具有性质 ${\cal P}$。我们证明，可以在多项式时间内判定 $M$ 是否可以被完成成一个无环有向图，该有向图具有从指定顶点 $s$ 出发的出分支，而判定 $M$ 是否存在一个无环定向，该定向同时具有出分支和入分支（即双极定向）是 NP 完全的。我们证明，判定 $M$ 是否可以被定向成包含两个指定顶点之间的有向路径是 NP 完全的。最后，我们描述了一个多项式算法，用于判定一个无环有向图 $D$ 是否存在一个出分支 $B^+_s$，使得有向图 $D-A(B^+_s)$（在底层意义下）是连通的。基于此，我们提出了一个开放问题：判定一个无环混合图的边是否可以被定向，使得结果是具有非分离出分支的无环有向图的复杂性。
英文摘要
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.
