Every edge-weighted planar graph with k terminals on f faces is claimed to admit an exact planar emulator of size O(f^2 k^2), interpolating known O(k^2) and O(k^4) bounds.
Shortcut partitions in minor-free graphs: Steiner point removal, distance oracles, tree covers, and more
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Paths and Intersections: Exact Emulators for Planar Graphs
Every edge-weighted planar graph with k terminals on f faces is claimed to admit an exact planar emulator of size O(f^2 k^2), interpolating known O(k^2) and O(k^4) bounds.