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arXiv · 2507.09620

Paths and Intersections: Exact Emulators for Planar Graphs

George Z. Li·Zihan Tan·Tianyi Zhang
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Abstract

We study vertex sparsification for preserving distances in planar graphs. Given an edge-weighted planar graph with $k$ terminals, the goal is to construct an emulator, which is a smaller edge-weighted planar graph that contains the terminals and exactly preserves the pairwise distances between them. We construct exact planar emulators of size $O(f^2k^2)$ in the setting where terminals lie on $f$ faces in the planar embedding of the input graph. Our result generalizes and interpolates between the previous results of Chang and Ophelders and Goranci, Henzinger, and Peng which is an $O(k^2)$ bound in the setting where all terminals lie on a single face (i.e., $f=1$), and the result of Krauthgamer, Nguyen, and Zondiner, which is an $O(k^4)$ bound for the general case (i.e., $f=k$). Our construction follows a recent new way of analyzing graph structures, by viewing graphs as paths and their intersections, which we believe is of independent interest.

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George Z. Li, Zihan Tan, Tianyi Zhang. 2025-07-13. Paths and Intersections: Exact Emulators for Planar Graphs. https://arxiv.org/abs/2507.09620

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