Heidi Steiger | Search arXiv

arXiv2024

Graph Skeletons and Diminishing Minors

We introduce the notion of coarse bottlenecking in graphs and coarse skeletons of graphs and show how bottlenecking guarantees that a skeleton resembles (up to quasi-isometry) the original graph. We show how these tools can be used to simplify the structure of graphs upto quasi-isometry that have an excluded asymptotic minor, reducing it to a skeleton of the original containing no $3$-fat minor. We give an example to show that a similar result does not hold for $2$-fat minors. This makes progress towards a Conjecture posed by Georgakopoulos and Papasoglu.

math.MG↗

arXiv2024

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 Erdős-Menger-type Conjecture, in the spirit of the Erdős-Menger conjecture which was proven after decades by Aharoni and Berger.

math.MG↗