Connected-homomorphism-homogeneous graphs

TL;DR

This paper classifies finite connected-homomorphism-homogeneous graphs, extending the concept of homogeneity by replacing isomorphisms with homomorphisms, and provides a hierarchy of these classes.

Contribution

It offers a classification of finite C-HH graphs and derives classifications for related classes, expanding understanding of homomorphism-based graph symmetries.

Findings

Classified finite C-HH graphs where homomorphisms extend to endomorphisms.

Derived classifications for C-HI and C-MI graphs from known C-II graphs.

Established a hierarchy for classes of connected-homomorphism-homogeneous graphs.

Abstract

A relational structure is (connected-)homogeneous if every isomorphism between finite (connected) substructures extends to an automorphism of the structure. We investigate notions which generalise (connected-)homogeneity, where "isomorphism" may be replaced by "homomorphism" or "monomorphism" in the definition. Specifically, we study the classes of finite connected-homomorphism-homogeneous graphs, with the aim of producing classifications. The main result is a classification of the finite C-HH graphs, where a graph G is C-HH if every homomorphism from a finite connected induced subgraph of G into G extends to an endomorphism of G. The finite C-II (connected-homogeneous) graphs were classified by Gardiner in 1976, and from this we obtain classifications of the finite C-HI and C-MI finite graphs. Although not all the classes of finite connected-homomorphism-homogeneous graphs are completely characterised, we may still obtain the final hierarchy picture for these classes.