Phylogenetic trees and homomorphisms

TL;DR

This paper explores the structure of homomorphism orders in graphs and relational structures, demonstrating their universality, gaps, and fractal properties, and applies these concepts to phylogenetic trees, revealing their combinatorial characteristics.

Contribution

It provides a comprehensive characterization of homomorphism orders, proves their universality and fractal properties, and connects these findings to the combinatorial analysis of phylogenetic trees.

Findings

Pairs of finite graphs forming gaps in homomorphism order characterized

Homomorphism order of directed graphs shown to be universal

Phylogenetic relation graphs are trees, related to underlying phylogenetic trees

Abstract

In Chapter 1 we fully characterise pairs of finite graphs which form a gap in the full homomorphism order. This leads to a simple proof of the existence of generalised duality pairs. We also discuss how such results can be carried to relational structures with unary and binary relations. In Chapter 2 we show a very simple and versatile argument based on divisibility which immediately yields the universality of the homomorphism order of directed graphs and discuss three applications. In chapter 3, we show that every interval in the homomorphism order of finite undirected graphs is either universal or a gap. Together with density and universality this "fractal" property contributes to the spectacular properties of the homomorphism order. In Chapter 4 we analyze the phylogenetic information content from a combinatorial point of view by considering the binary relation on the set of taxa defined by the existence of a single event separating two taxa. We show that the graph-representation of this relation must be a tree. Moreover, we characterize completely the relationship between the tree of such relations and the underlying phylogenetic tree.