发表机构
University of Leeds; Aix-Marseille Université（利兹大学; 艾克斯-马赛大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明：若两图强等距路径复杂度有界且存在良好根化的有界划分，则存在树宽不超过目标的图，使源图与其有加性拟等距，并解决开放问题。
AI 中文摘要
对于图$H$，$\langle H \rangle$表示$H$的所有细分图构成的类，$tw(H)$表示$H$的树宽。本文证明如下结论：对于$k\geq 1, R\geq 1$，设$G,H$是两个图，使得$G$和$\langle H \rangle$的强等距路径复杂度（Chakraborty等人[\textsc{Disc. Math., 2026}]）均至多为$k$，且$G$允许一个诚实的、“良好根化”的$R$-有界$H$-划分。则存在一个图$F$，满足$tw(F)\leq tw(H)$，使得$G$与$F$之间存在$(1,33\cdot R\cdot k^2)$-拟等距。利用Albrechtsen、Distel和Georgakopoulos（2025）的结果，我们还得到$K_{2,t}$-渐近无 minors 图允许到$K_{2,t}$-无 minors 图的具有加性扭曲的拟等距。这回答了上述作者提出的一个开放问题。作为证明的一部分，我们结合基于图划分的方法和基于分层划分的方法（Chepoi等人[\textsc{Discrete Comput. Geom.} 2012]），在源图和目标图的所有细分图都具有有界强等距路径复杂度的情况下，获得加性拟等距。
英文摘要
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.
