On the Homomorphism Order of Oriented Paths and Trees

TL;DR

This paper investigates the homomorphism order of oriented paths and trees, demonstrating that most intervals are universal except for those involving paths of height at most 3, which form a chain.

Contribution

It extends the understanding of homomorphism orders by characterizing universality of intervals within oriented paths and trees, identifying exceptions.

Findings

Intervals between oriented paths or trees of height ≥ 4 are universal.

Intervals involving paths of height ≤ 3 form a chain and are exceptions.

Most intervals in the homomorphism order of these classes are universal.

Abstract

A partial order is universal if it contains every countable partial order as a suborder. In 2017, Fiala, Hubi\v{c}ka, Long and Ne\v{s}et\v{r}il showed that every interval in the homomorphism order of graphs is universal, with the only exception being the trivial gap $[K_1,K_2]$. We consider the homomorphism order restricted to the class of oriented paths and trees. We show that every interval between two oriented paths or oriented trees of height at least 4 is universal. The exceptional intervals coincide for oriented paths and trees and are contained in the class of oriented paths of height at most 3, which forms a chain.