When can splits be drawn in the plane?

TL;DR

This paper characterizes which sets of splits can be represented by planar split networks, linking them to oriented matroids and line arrangements, and provides a simple description of maximal collections.

Contribution

It introduces a characterization of planar split networks using oriented matroids and line arrangements, and describes maximal collections of splits.

Findings

Characterization of split collections representable by planar networks

Connection between split networks and oriented matroids

Simple description of maximal split collections

Abstract

Split networks are a popular tool for the analysis and visualization of complex evolutionary histories. Every collection of splits (bipartitions) of a finite set can be represented by a split network. Here we characterize which collection of splits can be represented using a planar split network. Our main theorem links these collections of splits with oriented matroids and arrangements of lines separating points in the plane. As a consequence of our main theorem, we establish a particularly simple characterization of maximal collections of these splits.