AI 中文总结
本文证明图为拟平面图当且仅经两类有界迭代操作得到，推广了字符串图为拟平面图的结论，还推导了拟平面图的收缩子式性质及树分解相关结果，为拟等距等提供了工具。
AI 中文摘要
我们证明：图$G$是拟平面的，即与平面图拟等距，当且仅当可通过以下两种操作的有限次有界迭代得到：a) 将每条边细分为有界长度的路径；b) 取覆盖$G$的连通子图族的交图。该结论同时适用于无限图及具有一致常数的有限图族。反向推导依赖并推广了Davies的深层结果（Chang、Conroy、Tan和Zheng也独立部分证明了该结果），即每个字符串图都是拟平面的；正向推导则需要新的思路。作为证明的副产品，我们推导出拟平面图的每个收缩子式都是拟平面的。此外，若图$G$具有黏附集直径有界且诱导袋为拟平面的树分解，则$G$本身也是拟平面的。我们的结果也适用于其他图类，并提供了多种理解拟等距及图间双李普希茨等价性的工具。
英文摘要
We prove that a graph $G$ is quasi-planar - i.e. quasi-isometric to a planar graph - if and only if it can be obtained by iterating the following two operations a bounded number of times: a) subdividing each edge into a path of bounded length, and b) taking the intersection graph of a family of connected subgraphs covering $G$. This applies both to infinite graphs, and to families of finite graphs with uniform constants. The backward implication relies on, and generalises, a deep result of Davies, partly proved independently by Chang, Conroy, Tan & Zheng, saying that every string graph is quasi-planar. The forward implication requires new ideas. As a byproduct of our proofs, we deduce that every contraction minor of a quasi-planar graph is quasi-planar. Moreover, if $G$ admits a tree-decomposition with adhesions of bounded diameter and quasi-planar induced bags, then $G$ is itself quasi-planar. Our results apply to other graph classes as well, and we offer various tools for understanding quasi-isometries as well as bi-Lipschitz equivalences between graphs.
