Encoding and ordering X-cactuses

TL;DR

This paper introduces a novel encoding for X-cactuses, a type of phylogenetic network, and explores their structural properties and partial orderings to advance phylogenetic network analysis.

Contribution

It presents a new encoding method for X-cactuses using partitions of species set X and defines a partial order on these structures, extending phylogenetic tree properties.

Findings

Developed an encoding of X-cactuses via partitions of X

Defined a partial order on X-cactuses and analyzed its properties

Extended fundamental properties of phylogenetic trees to X-cactuses

Abstract

Phylogenetic networks are a generalization of evolutionary or phylogenetic trees that are commonly used to represent the evolution of species which cross with one another. A special type of phylogenetic network is an {\em $X$-cactus}, which is essentially a cactus graph in which all vertices with degree less than three are labelled by at least one element from a set $X$ of species. In this paper, we present a way to {\em encode} $X$-cactuses in terms of certain collections of partitions of $X$ that naturally arise from $X$-cactuses. Using this encoding, we also introduce a partial order on the set of $X$-cactuses (up to isomorphism), and derive some structural properties of the resulting partially ordered set. This includes an analysis of some properties of its least upper and greatest lower bounds. Our results not only extend some fundamental properties of phylogenetic trees to $X$-cactuses, but also provides a new approach to solving topical problems in phylogenetic network theory such as deriving consensus networks.