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Recently, there has been a growing interest in the relationships betweenunrooted and rooted phylogenetic networks. In this context, a natural questionto ask is if an unrooted phylogenetic network U can be oriented as a rootedphylogenetic network such that the latter satisfies certain structuralproperties. In a recent preprint, Bulteau et al. claim that it is computationalhard to decide if U has a funneled (resp. funneled tree-child) orientation, forwhen the internal vertices of U have degree at most 5. Unfortunately, the proofof their funneled tree-child result appears to be incorrect. In this paper, wepresent a corrected proof and show that hardness remains for other popularclasses of rooted phylogenetic networks such as funneled normal and funneledreticulation-visible. Additionally, our results hold regardless of whether U isrooted at an existing vertex or by subdividing an edge with the root.

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