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

Tropical hyperelliptic curves in the plane