  Isoradial embeddings of planar graphs play a crucial role in the study of
several models of statistical mechanics, such as the Ising and dimer models.
Kenyon and Schlenker give a combinatorial characterization of planar graphs
admitting an isoradial embedding, and describe the space of such embeddings. In
this paper we prove two results of the same type for generalizations of
isoradial embeddings: isoradial immersions and minimal immersions. We show that
a planar graph admits a flat isoradial immersion if and only if its
train-tracks do not form closed loops, and that a bipartite graph has a minimal
immersion if and only if it is minimal. In both cases we describe the space of
such immersions. The techniques used are different in both settings, and
distinct from those of Kenyon and Schlenker. We also give an application of our
results to the dimer model defined on bipartite graphs admitting minimal
immersions.
