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.

中文翻译：

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平面上的热带超椭圆曲线

摘要地，热带超椭圆曲线是度量图，允许对树进行二对一的谐波形态。它们也显示为平面中的嵌入式热带曲线，这是由内部所有晶格点共线的多边形的三角剖分引起的。我们证明了超椭圆图只能由这种多边形产生。在此过程中，我们将证明某些图形由于完全组合障碍而不会热带地嵌入平面中，无论其度量标准实际上是否为超椭圆形。