  Abstractly, tropical hyperelliptic curves are metric graphs that admit a
two-to-one harmonic morphism to a tree. They also appear as embedded tropical
curves in the plane arising from triangulations of polygons with all interior
lattice points collinear. We prove that hyperelliptic graphs can only arise
from such polygons. Along the way we will prove certain graphs do not embed
tropically in the plane due to entirely combinatorial obstructions, regardless
of whether their metric is actually hyperelliptic.
