The tropical crossing number of a finite graph

In 2015, Cartwright et al. showed that any $3$-regular metric graph arises as the skeleton of a tropical plane curve with nodes allowed. They introduced the tropical crossing number of a metric graph as the minimum number of nodes required for that graph with the prescribed lengths. We introduce the tropical crossing number of a finite, non-metric graph, the minimum number of nodes required to achieve that graph with any lengths on its edges. We prove that for any positive integer $d$ there exists a graph whose tropical crossing number is equal to $d$; moreover, this graph can be chosen with any prescribed graph-theoretic crossing number at most $d$. We then introduce and use computational methods to find the tropical crossing number of the smallest non-tropically planar graph, the lollipop graph of genus $3$. We also show that our tropical crossing number can grow quadratically in the number of vertices of the graph.