Constant-parameter scattering partitions for planar graphs

Determine whether planar graphs admit \((O(1),O(1))\)-scattering partitions.

Background

Scattering partitions require every shortest path to intersect only a bounded number of clusters, while also imposing strong diameter guarantees on the clusters. The paper notes that such partitions are known for bounded-doubling and pathwidth graphs, as well as chordal and series-parallel graphs, but the corresponding existence question for planar graphs remains unresolved.

References

It is still open whether planar graphs admit (O(1),O(1))-scattering partitions.

— Geometric Hashing in MPC with Applications to Clustering and Stochastic Embeddings  (2610.21371 - Bhore et al., 15 Oct 2026) in Section 1, subsection “Related work”