  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.
