On the existence of funneled orientations for classes of rooted phylogenetic networks

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 1/' 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 NP-hard to decide if 1/' has a funneled (resp. funneled tree-child) orientation, for when the internal vertices of 1/' have degree at most 5. Unfortunately, the proof of their funneled tree-child result appears to be incorrect. In this paper, we show that, despite their incorrect proof, it is NP-hard to decide if 1/' has a funneled tree-child orientation even if each internal vertex has degree 5 and that NP-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 1/' is rooted at an existing vertex or by subdividing an edge with the root.