Tropically planar graphs

Forbidden Patterns in Tropical Plane Curves

“…The following three technical results are extracted from the proof of [5,Theorem 3.4], where cut edges are called "bridges". Lemma 1.…”

“…Conversely, each unimodular regular triangulations of P arises in this way. In that case the edges of G (with their lengths, which we do not need here) describe T as a point in the moduli space M planar g of tropical plane curves of genus g. Skeleta of unimodular regular triangulations of lattice polygons are tropically planar or "troplanar" graphs in [5].…”

“…In the smooth case ∆ is a unimodular triangulation of P , and the planar graph G is trivalent of genus g, where g agrees with the number of interior lattice points of P . Such graphs have been called tropically planar by Coles et al [5]. Here we are concerned with the question: Which graphs G occur in this way?…”

“…For any fixed g ≥ 1, Castryck and Voight gave a procedure for finding finitely many lattice polygons with exactly g interior lattice points whose triangulations suffice to produce all such graphs [4]. This was employed in [2] and [5] to computationally determine the tropically planar graphs of genus g ≤ 7. Despite its success that approach is severely limited by the combinatorial explosion of the number of planar graphs and the number of triangulations of the relevant polygons.…”

“…While it is doubtful that this can be pushed much further, here we seek to find obstructions for a given trivalent graph to be realizable as a tropical plane curve. It was known previously that graphs with a sprawling node [3,Proposition 4.1], crowded graphs [12,Lemma 3.5] and TIE-fighter graphs [5,Theorem 3.4] cannot occur; cf. Figure 1.…”