  We expand upon the notion of bottlenecking introduced in our earlier work,
characterizing a spectrum of graphs and showing that this naturally extends to
a concept of coarse bottlenecking. We examine how bottlenecking differs from
other notions of connectedness. We show how the notion of bottlenecking
provides a different approach to coarsening measures of connectedness than the
Coarse Menger Conjecture proposed independently by Georgakopoulos and Papasoglu
as well as Albrechtsen, Huynh, Jacobs, Knappe, and Wollan, which was recently
disproved by a counterexample. We give a proof of a coarse Menger-type theorem
for the class of coarsely bottlenecked graphs, and provide a conjecture that
would extend this to all graphs.
