Unique Least Common Ancestors and Clusters in Directed Acyclic Graphs

TL;DR

This paper explores the relationship between clusters and least common ancestors in DAGs, focusing on those with unique LCAs for certain subsets, relevant for phylogenetic network modeling.

Contribution

It establishes a connection between pre-k-ary clustering systems and DAGs with unique LCAs, introducing new characterizations of these structures.

Findings

Pre-k-ary clustering systems correspond to DAGs with unique LCAs.

k-ary T-systems relate to DAGs with stronger LCA uniqueness conditions.

The study links canonical transit functions to closure functions in set systems.

Abstract

We investigate the connections between clusters and least common ancestors (LCAs) in directed acyclic graphs (DAGs). We focus on the class of DAGs having unique least common ancestors for certain subsets of their minimal elements since these are of interest, particularly as models of phylogenetic networks. Here, we use the close connection between the canonical k-ary transit function and the closure function on a set system to show that pre-k-ary clustering systems are exactly those that derive from a class of DAGs with unique LCAs. Moreover, we show that k-ary T-systems and k-weak hierarchies are associated with DAGs that satisfy stronger conditions on the existence of unique LCAs for sets of size at most k.