Clustering systems of phylogenetic networks

Abstract

AbstractRooted 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 $${{\,\mathrm{\texttt{C}}\,}}(v)$$ C ( v ) of descendant taxa of a vertex v. The clustering system $$\mathscr {C}_N$$ 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 $$\mathscr {C}_T$$ C T . Although this is no longer true for networks in general, it is of interest to relate properties of N and $$\mathscr {C}_N$$ 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 systems of the following form: If N is a network of type $$\mathbb {X}$$ X , then $$\mathscr {C}_N$$ C N satisfies $$\mathbb {Y}$$ Y , and conversely if $$\mathscr {C}$$ C is a clustering system satisfying $$\mathbb {Y},$$ Y , then there is network N of type $$\mathbb {X}$$ X such that $$\mathscr {C}\subseteq \mathscr {C}_N$$ C ⊆ C N .This, in turn, allows us to investigate the mutual dependencies between the distinct types of networks in much detail.