Phylogenetic networks generalize phylogenetic trees by representing reticulate evolution. Tree-based networks and their support trees have been extensively studied, but not all networks are tree-based. To measure how far such networks are from being tree-based, Suzuki and Hayamizu (2025) formulated the problem of finding the support network with minimum level of a given rooted almost-binary phylogenetic network. They conjectured that this problem is NP-hard and provided exponential-time algorithms. In this paper, we prove this conjecture by showing that, for every fixed integer $k \geq 1$, it is NP-complete to decide whether the minimum level is at most $k$.

翻译：系统发育网络通过代表网状进化来概括系统发育树。基于树的网络及其支持树已被广泛研究，但并非所有网络都是基于树的。为了衡量这些网络与基于树的网络的偏离程度，Suzuki和Hayamizu（2025）提出了寻找给定有根几乎二元系统发育网络的最小水平支持网络的问题。他们推测该问题是NP难的，并提供了指数时间算法。在本文中，我们通过证明对于每个固定整数$k \geq 1$，判断最小水平是否至多为$k$是NP完全的，从而证实了这一猜想。