We study the moduli space of metric graphs that arise from tropical plane
curves. There are far fewer such graphs than tropicalizations of classical
plane curves. For fixed genus g, our moduli space is a stacky fan whose cones
are indexed by regular unimodular triangulations of Newton polygons with g
interior lattice points. It has dimension 2g+1 unless g≤3 or g=7.
We compute these spaces explicitly for g≤5.Comment: 31 pages, 25 figure