Bijections for Ranked Tree-Child Networks

TL;DR

This paper provides bijective proofs for key counting results of ranked tree-child networks, a class of phylogenetic networks, addressing open questions and enhancing understanding of their combinatorial structure.

Contribution

It introduces bijective proofs for existing counting results and resolves open questions from prior research on ranked tree-child networks.

Findings

Bijections for three counting results established

Answers to two open questions from previous work

Enhanced understanding of the combinatorial structure of ranked tree-child networks

Abstract

The class of ranked tree-child networks, tree-child networks arising from an evolution process with a fixed embedding into the plane, has recently been introduced by Bienvenu, Lambert, and Steel. These authors derived counting results for this class. In this note, we will give bijective proofs of three of their results. Two of our bijections answer questions raised in their paper.