  We investigate data-enriched models, like Petri nets with data, where
executability of a transition is conditioned by a relation between data values
involved. Decidability status of various decision problems in such models may
depend on the structure of data domain. According to the WQO Dichotomy
Conjecture, if a data domain is homogeneous then it either exhibits a well
quasi-order (in which case decidability follows by standard arguments), or
essentially all the decision problems are undecidable for Petri nets over that
data domain. We confirm the conjecture for data domains being 3-graphs (graphs
with 2-colored edges). On the technical level, this results is a significant
step beyond known classification results for homogeneous structures.
