O(1)-Distortion Planar Emulators for String Graphs

We show that every unweighted string graph $G$ has an $O(1)$-distortion planar emulator: that is, there exists an (edge-weighted) planar graph $H$ with $V(H)=V(G)$, such that every pair of vertices $(u,v)$ satisfies ${\delta }_G(u,v)\le {\delta }_H(u,v)\le O(1)\cdot {\delta }_G(u,v)\mathrm{.}$