Paths and Intersections: Exact Emulators for Planar Graphs

TL;DR

This paper introduces a method for creating exact, smaller planar graph emulators that preserve distances between terminals, generalizing previous bounds based on the number of faces containing terminals.

Contribution

It presents a new construction for exact planar emulators with size bounds depending on the number of faces with terminals, extending previous results for special cases.

Findings

Constructed exact planar emulators of size $O(f^2k^2)$.

Generalizes previous bounds for terminal placement.

Introduces a new graph analysis technique based on paths and intersections.

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.