Additive Quasi-isometry via rooted graph partitions and layering partition

Abstract

For a graph $H$, $⟨H⟩$ denotes the class of all subdivisions of $H$ and $tw(H)$ denotes the treewidth of $H$. In this paper, we prove the following. For $k\ge 1,R\ge 1$, let $G,H$ be two graphs such that strong isometric path complexities (Chakraborty et al. [\textsc{Disc. Math., 2026}]) of both $G$ and $⟨H⟩$ are at most $k$, and $G$ admits an honest, ``nicely rooted'' $R$-bounded $H$-partition. Then, there is a graph $F$ with $tw(F)\le tw(H)$ such that $G$ admits a $(1,33\cdot R\cdot k^2)$-quasi-isometry to $F$. Using results of Albrechtsen, Distel, and Georgakopoulos (2025), we also obtain that $K_{2,t}$-asymptotic minor-free graphs admit quasi-isometries with additive distortion to $K_{2,t}$-minor-free graphs. This answers an open question raised by the above authors. As part of our proof, we combine the graph-partition based method and the layering partition based method (Chepoi et al. [\textsc{Discrete Comput. Geom.} 2012]) to obtain additive quasi-isometry when both the source and all subdivisions of the target graph have bounded strong isometric path complexity.