  We show that for most pairs of surfaces, there exists a finite subgraph of
the flip graph of the first surface so that any injective homomorphism of this
finite subgraph into the flip graph of the second surface can be extended
uniquely to an injective homomorphism between the two flip graphs. Combined
with a result of Aramayona-Koberda-Parlier, this implies that any such
injective homomorphism of this finite set is induced by an embedding of the
surfaces. We also include images of several flip graphs in an appendix.
