Coarse-minor exclusion and quasi-isometric resemblance to minor-free graphs

Establish that if a graph excludes a fixed graph as a fat minor, then its metric structure is quasi-isometric to that of a graph excluding a classic minor.

Background

The paper places its results within the broader program of coarse graph theory, where fat minors serve as metric analogues of ordinary graph minors and quasi-isometries measure coarse resemblance between graphs. The stated conjecture proposes that excluding a fixed graph as a fat minor should force a graph to have, up to quasi-isometry, the structure of a graph excluding a corresponding classic minor.

The conjecture is attributed to Georgakopoulos and Papasoglu and is presented as the overarching motivation for the systematic development of coarse counterparts to structural graph theory. The paper does not resolve this conjecture; instead, it proves a coarse analogue of a theorem about graphs whose vertices are covered by a bounded number of shortest paths.

References

Recently, Georgakopoulous and Papasoglu launched a systematic investigation of the coarse counterpart of the theory of Graph Minors, with the following overarching conjecture in mind: If a graph $G$ excludes some fixed graph as a fat minor, then the metric structure of $G$ resembles that of a graph excluding a classic minor (there are subtleties regarding this statement, see the discussion in).

— On graphs coverable by chubby shortest paths  (2503.02160 - Hatzel et al., 4 Mar 2025) in Section 1, Introduction