Graph Skeletons and Diminishing Minors

We define coarse skeletons of graphs in terms of two constants. We introduce the notion of coarse bottlenecking in graphs and show how it can guarantee that a skeleton resembles (up to quasi-isometry) the original graph. We show how notions similar to the coarse skeleton have been previously used to classify some coarse families of graphs. We explore the properties of a coarse skeleton and of combinations of skeletons. We show how these tools can be used to simplify the structure of graphs that have an excluded asymptotic minor, reducing it to a skeleton of the original containing at most a 3-fat minor, we give an example to show that a similar result does not hold for 2-fat minors.