Title: Tree-based networks: characterisations, metrics, and support trees,
Abstract: Phylogenetic networks generalise phylogenetic trees and allow for the
accurate representation of the evolutionary history of a set of present-day
species whose past includes reticulate events such as hybridisation and lateral
gene transfer. One way to obtain such a network is by starting with a (rooted)
phylogenetic tree $T$, called a base tree, and adding arcs between arcs of $T$.
The class of phylogenetic networks that can be obtained in this way is called
tree-based networks and includes the prominent classes of tree-child and
reticulation-visible networks. Initially defined for binary phylogenetic
networks, tree-based networks naturally extend to arbitrary phylogenetic
networks. In this paper, we generalise recent tree-based characterisations and
associated proximity measures for binary phylogenetic networks to arbitrary
phylogenetic networks. These characterisations are in terms of matchings in
bipartite graphs, path partitions, and antichains. Some of the generalisations
are straightforward to establish using the original approach, while others
require a very different approach. Furthermore, for an arbitrary tree-based
network $N$, we characterise the support trees of $N$, that is, the tree-based
embeddings of $N$. We use this characterisation to give an explicit formula for
the number of support trees of $N$ when $N$ is binary. This formula is written
in terms of the components of a bipartite graph.