CSD Homomorphisms Between Phylogenetic Networks

Since Darwin, species trees have been used as a simplified description of the relationships which summarize the complicated network $N$ of reality. Recent evidence of hybridization and lateral gene transfer, however, suggest that there are situations where trees are inadequate. Consequently it is important to determine properties that characterize networks closely related to $N$ and possibly more complicated than trees but lacking the full complexity of $N$. A connected surjective digraph map (CSD) is a map $f$ from one network $N$ to another network $M$ such that every arc is either collapsed to a single vertex or is taken to an arc, such that $f$ is surjective, and such that the inverse image of a vertex is always connected. CSD maps are shown to behave well under composition. It is proved that if there is a CSD map from $N$ to $M$, then there is a way to lift an undirected version of $M$ into $N$, often with added resolution. A CSD map from $N$ to $M$ puts strong constraints on $N$. In general, it may be useful to study classes of networks such that, for any $N$, there exists a CSD map from $N$ to some standard member of that class.Comment: 19 pages, 3 figure

Restricted trees: simplifying networks with bottlenecks

Suppose N is a phylogenetic network indicating a complicated relationship among individuals and taxa. Often of interest is a much simpler network, for example, a species tree T, that summarizes the most fundamental relationships. The meaning of a species tree is made more complicated by the recent discovery of the importance of hybridizations and lateral gene transfers. Hence it is desirable to describe uniform well-defined procedures that yield a tree given a network N. A useful tool toward this end is a connected surjective digraph (CSD) map f from N to N' where N' is generally a much simpler network than N. A set W of vertices in N is "restricted" if there is at most one vertex from which there is an arc into W, thus yielding a bottleneck in N. A CSD map f from N to N' is "restricted" if the inverse image of each vertex in N' is restricted in N. This paper describes a uniform procedure that, given a network N, yields a well-defined tree called the "restricted tree" of N. There is a restricted CSD map from N to the restricted tree. Many relationships in the tree can be proved to appear also in N.Comment: 17 pages, 2 figure

Folding and unfolding phylogenetic trees and networks

Phylogenetic networks are rooted, labelled directed acyclic graphs which are commonly used to represent reticulate evolution. There is a close relationship between phylogenetic networks and multi-labelled trees (MUL-trees). Indeed, any phylogenetic network $N$ can be "unfolded" to obtain a MUL-tree $U(N)$ and, conversely, a MUL-tree $T$ can in certain circumstances be "folded" to obtain a phylogenetic network $F(T)$ that exhibits $T$. In this paper, we study properties of the operations $U$ and $F$ in more detail. In particular, we introduce the class of stable networks, phylogenetic networks $N$ for which $F(U(N))$ is isomorphic to $N$, characterise such networks, and show that they are related to the well-known class of tree-sibling networks.We also explore how the concept of displaying a tree in a network $N$ can be related to displaying the tree in the MUL-tree $U(N)$. To do this, we develop a phylogenetic analogue of graph fibrations. This allows us to view $U(N)$ as the analogue of the universal cover of a digraph, and to establish a close connection between displaying trees in $U(N)$ and reconcilingphylogenetic trees with networks

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