Recently, there has been a growing interest in the relationships between unrooted and rooted phylogenetic networks. In this context, a natural question to ask is if an unrooted phylogenetic network U can be oriented as a rooted phylogenetic network such that the latter satisfies certain structural properties. In a recent preprint, Bulteau et al. claim that it is computational hard to decide if U has a funneled (resp. funneled tree-child) orientation, for when the internal vertices of U have degree at most 5. Unfortunately, the proof of their funneled tree-child result appears to be incorrect. In this paper, we present a corrected proof and show that hardness remains for other popular classes of rooted phylogenetic networks such as funneled normal and funneled reticulation-visible. Additionally, our results hold regardless of whether U is rooted at an existing vertex or by subdividing an edge with the root.

翻译：近年来，无根与有根系统发育网络之间的关系日益受到关注。在此背景下，一个自然的问题是：能否将无根系统发育网络U定向为满足特定结构性质的有根系统发育网络？在最近的一篇预印本中，Bulteau等人声称，当U的内部顶点度数不超过5时，判定U是否存在漏斗定向（或漏斗树子定向）是计算困难的。遗憾的是，他们关于漏斗树子定向的证明似乎存在错误。本文给出了修正后的证明，并表明该困难性对其他常见的有根系统发育网络类别（如漏斗正规网络和漏斗网状可见网络）依然成立。此外，无论U是以现有顶点为根，还是通过将根与某条边的细分点相结合，我们的结果均成立。