Clustering Systems of Phylogenetic Networks
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Rooted acyclic graphs appear naturally when the phylogenetic relationship of a set X of taxa involves not only speciations but also recombination, horizontal transfer, or hybridization, that cannot be captured by trees. A variety of classes of such networks have been discussed in the literature, including phylogenetic, level-1, tree-child, tree-based, galled tree, regular, or normal networks as models of different types of evolutionary processes. Clusters arise in models of phylogeny as the sets 𝙲(v) of descendant taxa of a vertex v. The clustering system 𝒞_N comprising the clusters of a network N conveys key information on N itself. In the special case of rooted phylogenetic trees, T is uniquely determined by its clustering system 𝒞_T. Although this is no longer true for networks in general, it is of interest to relate properties of N and 𝒞_N. Here, we systematically investigate the relationships of several well-studied classes of networks and their clustering systems. The main results are correspondences of classes of networks and clustering system of the following form: If N is a network of type 𝕏, then 𝒞_N satisfies 𝕐, and conversely if 𝒞 is a clustering system satisfying 𝕐 then there is network N of type 𝕏 such that 𝒞⊆𝒞_N.This, in turn, allows us to investigate the mutual dependencies between the distinct types of networks in much detail.
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