Asymptotic structure. II. Path-width and additive quasi-isometry

TL;DR

The paper proves that if a graph $G$ is $(L,C)$-quasi-isometric to a graph $H$ of bounded path-width, one can assign a finite edge-weighting $w$ to $H$ so that the same map $\phi$ becomes a $(1,C')$-quasi-isometry from $G$ to the weighted graph $(H,w)$, with $C'$ depending only on $L$, $C$, and the path-width bound. It develops a two-stage approach: (i) a local weighting near the image of a geodesic $P$ in $G$ that yields a $w$-geodesic $Q$ in $H$ and aligns $\phi(P)$ with $Q$, and (ii) a global extension of this weighting to all of $H$ via an inductive scheme on width using an additive bounder for a hereditary class. The second half of the argument handles infinite graphs by introducing a spanning geodesic framework that replaces a genuine geodesic when necessary, ensuring the global quasi-isometry to $(H,w)$ remains controlled. Altogether, the results bound the additive distortion in transferring quasi-isometric information to a weighted target, extending the scope of 1-Lipschitz quasi-isometries to wider graph classes, and clarifying when path-width (and line-width) constraints suffice to enable such a transformation.

Abstract

We show that if a graph $G$ admits a quasi-isometry $φ$ to a graph $H$ of bounded path-width, then we can assign a non-negative integer length to each edge of $H$, such that the same function $φ$ is a quasi-isometry to this weighted version of $H$, with error only an additive constant.