Tree-based unrooted phylogenetic networks

Phylogenetic networks are a generalization of phylogenetic trees that are used to represent non-tree-like evolutionary histories that arise in organisms such as plants and bacteria, or uncertainty in evolutionary histories. An \emph{unrooted} phylogenetic network on a nonempty, finite set $X$ of taxa, or \emph{network}, is a connected graph in which every vertex has degree 1 or 3 and whose leaf-set is $X$. It is called a \emph{phylogenetic tree} if the underlying graph is a tree. In this paper we consider properties of \emph{tree-based networks}, that is, networks that can be constructed by adding edges into a phylogenetic tree. We show that although they have some properties in common with their rooted analogues which have recently drawn much attention in the literature, they have some striking differences in terms of both their structural and computational properties. We expect that our results could eventually have applications to, for example, detecting horizontal gene transfer or hyrbridization which are important factors in the evolution of many organisms.