Additive Quasi-isometry via rooted graph partitions and layering partition

2026-09-28

AI Summary

This study addresses the long-standing challenge of constructing additive quasi-isometries for graphs with bounded strong isometric path complexity. To this end, the proposed approach innovatively integrates rooted partitioning and hierarchical decomposition techniques with treewidth theory, enabling rigorous algorithmic design and mathematical derivation to construct additive quasi-isometries for target graphs of low treewidth under specific complexity constraints. The primary contribution lies in proving the existence of such mappings while achieving controllable additive distortion. This result resolves a significant open problem concerning $K_{2,t}$-minor-free asymptotic graphs, thereby establishing a novel paradigm for the approximate embedding of graph structures.

Abstract

For a graph $H$, $\langle H \rangle$ denotes the class of all subdivisions of $H$ and $tw(H)$ denotes the treewidth of $H$. In this paper, we prove the following. For $k\geq 1, R\geq 1$, let $G,H$ be two graphs such that strong isometric path complexities (Chakraborty et al. [\textsc{Disc. Math., 2026}]) of both $G$ and $\langle H \rangle$ are at most $k$, and $G$ admits an honest, ``nicely rooted'' $R$-bounded $H$-partition. Then, there is a graph $F$ with $tw(F)\leq tw(H)$ 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.

Problem

• additive quasi-isometry
• strong isometric path complexity
• graph partition
• layering partition
• asymptotic minor-free graphs

Innovation

• Additive Quasi-isometry
• Graph Partitions
• Layering Partition
• Strong Isometric Path Complexity
• Asymptotic Minor-free Graphs