Orchard Networks are Trees with Additional Horizontal Arcs

TL;DR

This paper characterizes orchard phylogenetic networks as trees with horizontal arcs, linking them to tree-based networks, and demonstrates their connectivity under rNNI moves with a specific diameter bound.

Contribution

It introduces a new interpretation of orchard networks as trees with horizontal arcs and analyzes their structural properties and connectivity.

Findings

Orchard networks can be viewed as trees with horizontal arcs.

The space of orchard networks is connected under rNNI moves.

The diameter of this space is bounded by a specific function.

Abstract

Phylogenetic networks are used in biology to represent evolutionary histories. The class of orchard phylogenetic networks was recently introduced for their computational benefits, without any biological justification. Here, we show that orchard networks can be interpreted as trees with additional \emph{horizontal} arcs. Therefore, they are closely related to tree-based networks, where the difference is that in tree-based networks the additional arcs do not need to be horizontal. Then, we use this new characterization to show that the space of orchard networks is connected under the rNNI rearrangement move, with a diameter of at most $4kn+n\lceil \log_2(n) \rceil +2k+6n-8$.