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The scramble number of a graph is an invariant recently developed to aid in the study of divisorial gonality. In this paper we establish important properties of scramble number, showing that it is monotone under taking immersion minors and finding the minimal forbidden immersion minors for graphs of scramble number at most 2. We then prove that scramble number is NP-hard to compute, also providing a proof that computing gonality is NP-hard even for simple graphs. We also provide general lower bounds the scramble number of a Cartesian product of graphs, and apply these to compute gonality for many new families of product graphs.

Given a lattice polygon P with g interior lattice points, we associate to it the moduli space of tropical curves of genus g with Newton polygon P . We completely classify the possible dimensions such a moduli space can have. For non-hyperelliptic polygons the dimension must be between g and 2g + 1, and can take on any integer value in this range, with exceptions only in the cases of genus 3, 4, and 7. We provide a similar result for hyperelliptic polygons, for which the range of dimensions is from g to 2g − 1. In the case of non-hyperelliptic polygons, our results also hold for the moduli space of algebraic curves that are non-degenerate with respect to P .