  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.
