Phylogenetic Diversity (PD) is a prominent quantitative measure of the biodiversity of a collection of present-day species (taxa). This measure is based on the evolutionary distance among the species in the collection. Loosely speaking, if $mathcal{T}$ is a rooted phylogenetic tree whose leaf set $X$ represents a set of species and whose edges have real-valued lengths (weights), then the PD score of a subset $S$ of $X$ is the sum of the weights of the edges of the minimal subtree of $mathcal{T}$ connecting the species in $S$. In this paper, we define several natural variants of the PD score for a subset of taxa which are related by a known rooted phylogenetic network. Under these variants, we explore, for a positive integer $k$, the computational complexity of determining the maximum PD score over all subsets of taxa of size $k$ when the input is restricted to different classes of rooted phylogenetic networks
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