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~\cite{ChangO20} and Goranci, Henzinger, and Peng~\cite{goranci2020improved} 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~\cite{krauthgamer2014preserving}, which is an upper bound of $O(k^4)$ 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.