Bottlenecking in graphs and a coarse Menger-type theorem

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 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 formulate and prove a Coarse Menger-type theorem, and also propose a coarse Erd\H{o}s-Menger-type Conjecture, in the spirit of the Erd\H{o}s-Menger conjecture which was proven after decades by Aharoni and Berger.