# Two Fra\"iss\'e-style theorems for homomorphism-homogeneous relational structures

## TL;DR

This paper establishes two Fraïssé-style theorems for various notions of homomorphism-homogeneity in relational structures, advancing understanding of their existence, uniqueness, and classification, especially for countable graphs.

## Contribution

It proves two key theorems for twelve notions of homomorphism-homogeneity and clarifies their implications for the classification of countable homogeneous graphs.

## Key findings

- Complete classification of homomorphism-homogeneous undirected graphs
- Two Fraïssé-style theorems for twelve notions of homomorphism-homogeneity
- Insights into directed graph cases

## Abstract

In this paper, we state and prove two Fra\"{i}ss\'{e}-style results that cover existence and uniqueness properties for twelve of the eighteen different notions of homomorphism-homogeneity as introduced by Lockett and Truss, and provide forward directions and implications for the remaining six cases. Following these results, we completely determine the extent to which the countable homogeneous undirected graphs (as classified by Lachlan and Woodrow) are homomorphism-homogeneous; we also provide some insight into the directed graph case.