This paper studies the relationship between undirected (unrooted) and directed (rooted) phylogenetic networks. We describe a polynomial-time algorithm for deciding whether an undirected nonbinary phylogenetic network, given the locations of the root and reticulation vertices, can be oriented as a directed nonbinary phylogenetic network. Moreover, we characterize when this is possible and show that, in such instances, the resulting directed nonbinary phylogenetic network is unique. In addition, without being given the location of the root and the reticulation vertices, we describe an algorithm for deciding whether an undirected binary phylogenetic network $N$ can be oriented as a directed binary phylogenetic network of a certain class. The algorithm is fixed-parameter tractable (FPT) when the parameter is the level of $N$ and is applicable to classes of directed phylogenetic networks that satisfy certain conditions. As an example, we show that the well-studied class of binary tree-child networks satisfies these conditions.

翻译：本文研究了无根（无向）与有根（有向）系统发育网络之间的关系。我们描述了一种多项式时间算法，用于判定给定根节点和网状节点位置的无根非二叉树系统发育网络能否被定向为有根非二叉树系统发育网络。此外，我们刻画了这种定向可行的条件，并证明在此类情况下，生成的有根非二叉树系统发育网络具有唯一性。进一步地，在未提供根节点与网状节点位置的情况下，我们提出了一种算法，用于判定无根二叉树系统发育网络$N$能否被定向为特定类别的有根二叉树系统发育网络。当以$N$的层级作为参数时，该算法是固定参数可处理的（FPT），且适用于满足特定条件的有根系统发育网络类别。以一类得到广泛研究的二叉树子网络为例，我们证明该类网络满足上述条件。