On well quasi-order of graph classes under homomorphic image orderings

TL;DR

This paper investigates the well quasi-ordering of various graph classes under homomorphic image orderings, providing complete characterizations in most cases and highlighting open problems.

Contribution

It offers a comprehensive analysis of well quasi-ordering for classes defined by a single obstruction under homomorphic image orderings, extending previous work.

Findings

Complete characterizations for graphs, digraphs, and tournaments under standard ordering.

Identification of open questions for graphs under strong ordering.

Extension of homomorphic image ordering theory to well quasi-order analysis.

Abstract

In this paper we consider the question of well quasi-order for classes defined by a single obstruction within the classes of all graphs, digraphs and tournaments, under the homomorphic image ordering (in both its standard and strong forms). The homomorphic image ordering was introduced by the authors in a previous paper and corresponds to the existence of a surjective homomorphism between two structures. We obtain complete characterizations in all cases except for graphs under the strong ordering, where some open questions remain.