O(1)-Distortion Planar Emulators for String Graphs

TL;DR

This paper proves that every unweighted string graph can be approximated by a planar graph with constant distortion, enabling efficient planar representations of complex string graphs.

Contribution

It introduces the first construction of O(1)-distortion planar emulators for all unweighted string graphs, bridging string graph complexity and planar graph approximations.

Findings

Existence of O(1)-distortion planar emulators for all unweighted string graphs

Planar emulators preserve distances within constant factors

Facilitates efficient algorithms for string graphs using planar structures

Abstract

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 $\delta_G(u,v) \le \delta_H(u,v) \le O(1) \cdot \delta_G(u,v).$