New Characterisations of Tree-Based Networks and Proximity Measures

TL;DR

This paper introduces new characterisations of tree-based phylogenetic networks using path partitions, antichains, and bipartite matchings, and defines indices to measure their deviation from being tree-based.

Contribution

It provides novel characterisations of tree-based networks and efficient methods to compute indices quantifying their deviation from tree-based structures.

Findings

New characterisations using path partitions and matchings

A bipartite graph-based method for characterising tree-based networks

Efficient computation of deviation indices using maximum matchings

Abstract

Phylogenetic networks are a type of directed acyclic graph that represent how a set $X$ of present-day species are descended from a common ancestor by processes of speciation and reticulate evolution. In the absence of reticulate evolution, such networks are simply phylogenetic (evolutionary) trees. Moreover, phylogenetic networks that are not trees can sometimes be represented as phylogenetic trees with additional directed edges placed between their edges. Such networks are called {\em tree based}, and the class of phylogenetic networks that are tree based has recently been characterised. In this paper, we establish a number of new characterisations of tree-based networks in terms of path partitions and antichains (in the spirit of Dilworth's theorem), as well as via matchings in a bipartite graph. We also show that a temporal network is tree based if and only if it satisfies an antichain-to-leaf condition. In the second part of the paper, we define three indices that measure the extent to which an arbitrary phylogenetic network deviates from being tree based. We describe how these three indices can be described exactly and computed efficiently using classical results concerning maximum-sized matchings in bipartite graphs.