We prove that for every countable string graph $S$, there is a planar graph $G$ with $V(G)=V(S)$ such that \[ \frac{1}{23660800}d_S(u,v) \le d_G(u,v) \le 162 d_S(u,v) \] for all $u,v\in V(S)$, where $d_S(u,v)$, $d_G(u,v)$ denotes the distance between $u$ and $v$ in $S$ and $G$ respectively. In other words, string graphs are quasi-isometric to planar graphs. This theorem lifts a number of theorems from planar graphs to string graphs, we give some examples. String graphs have Assouad-Nagata (and asymptotic dimension) at most 2. Connected, locally finite, quasi-transitive string graphs are accessible. A finitely generated group $Γ$ is virtually a free product of free and surface groups if and only if $Γ$ is quasi-isometric to a string graph. Two further corollaries are that countable planar metric graphs and complete Riemannian planes are also quasi-isometric to planar graphs, which answers a question of Georgakopoulos and Papasoglu. For finite string graphs and planar metric graphs, our proofs yield polynomial time (for string graphs, this is in terms of the size of a representation given in the input) algorithms for generating such quasi-isometric planar graphs. We further extend our techniques to show that every complete Riemannian surfaces $Σ$ of bounded Euler genus has a triangulation $G\subset Σ$ such that $G^{(1)} \hookrightarrow Σ$ is a quasi-isometry, where $G^{(1)}$ is the simplicial 1-skeleton of $G$.

翻译：我们证明：对于每个可数弦图 $S$，存在一个平面图 $G$ 且 $V(G)=V(S)$，使得对所有 $u,v\in V(S)$ 有 \[ \frac{1}{23660800}d_S(u,v) \le d_G(u,v) \le 162 d_S(u,v) \]，其中 $d_S(u,v)$ 和 $d_G(u,v)$ 分别表示 $u$ 与 $v$ 在 $S$ 和 $G$ 中的距离。换言之，弦图与平面图是拟等距的。该定理将平面图的若干定理提升至弦图，我们给出了一些实例。弦图的 Assouad-Nagata 维数（以及渐近维数）至多为 2。连通的、局部有限的、拟传递弦图是可接近的。一个有限生成群 $\Gamma$ 是自由群与曲面群的自由积的有限扩张，当且仅当 $\Gamma$ 与某个弦图拟等距。另外两个推论是：可数平面度量图与完备黎曼平面也与平面图拟等距，这回答了 Georgakopoulos 与 Papasoglu 的一个问题。对于有限弦图与平面度量图，我们的证明给出了生成此类拟等距平面图的多项式时间算法（对弦图而言，复杂度与输入中给出的表示的规模相关）。我们进一步推广该技巧，证明了每个具有有界 Euler 亏格的完备黎曼曲面 $\Sigma$ 存在一个三角剖分 $G\subset \Sigma$，使得 $G^{(1)} \hookrightarrow \Sigma$ 是一个拟等距，其中 $G^{(1)}$ 是 $G$ 的单纯 1-骨架。