The scramble number of a graph provides a lower bound for gonality and an upper bound for treewidth, making it a graph invariant of interest. In this paper we study graphs of scramble number at most two, and give a classification of all such graphs with a finite list of forbidden topological minors. We then prove that there exists no finite list of forbidden topological minors to characterize graphs with scramble number at most k for any fixed k≥3.

The cop throttling number of a graph, introduced in 2018 by Breen et al., optimizes the balance between the number of cops used and the number of rounds required to catch the robber in a game of Cops and Robbers. In 2019, Cox and Sanaei studied a variant of Cops and Robbers in which the robber tries to occupy (or damage) as many vertices as possible and the cop tries to minimize this damage. They investigated the minimum number of vertices damaged by the robber over all games played on a given graph G, called the damage number of G. We introduce the natural parameter called the damage throttling number of a graph, denoted th(d)(G), which optimizes the balance between the number of cops used and the number of vertices damaged in the graph. We show that damage throttling and cop throttling share many properties, yet they exhibit interesting differences. We prove that th(d)(G) is tightly bounded above by one less than the cop throttling number. We discuss infinite families of graphs which attain equality for this bound, as well as graphs which have a greater gap between the damage throttling number and the cop throttling number. For most families of connected graphs G of order n that we consider in this paper, we prove that th(d)(G) = O(root n). However, we also find an infinite family of connected graphs G of order n for which th(d)(G) = Omega(n(2/3)).