Morphism extension classes of countable $L$-colored graphs

TL;DR

This paper investigates the hierarchy of morphism extension classes in countably infinite $L$-colored graphs, establishing that the classes coincide precisely when $L$ is a linear order, extending prior finite case results.

Contribution

It extends the study of morphism extension classes from finite to countably infinite $L$-colored graphs, providing a complete characterization of when these classes coincide.

Findings

$MH_L=HH_L$ if and only if $L$ is a linear order

Established the hierarchy for countably infinite $L$-colored graphs

Extended finite case results to infinite graphs

Abstract

In~\cite{Hartman:2014}, Hartman, Hubi\v cka and Ma\v sulovi\'c studied the hierarchy of morphism extension classes for finite $L$-colored graphs, that is, undirected graphs without loops where sets of colors selected from $L$ are assigned to vertices and edges. They proved that when $L$ is a linear order, the classes $MH_L$ and $HH_L$ coincide, and the same is true for vertex-uniform finite $L$-colored graphs when $L$ is a diamond. In this paper, we explore the same question for countably infinite $L$-colored graphs. We prove that $MH_L=HH_L$ if and only if $L$ is a linear order.