Uniqueness of maximum planar five-distance sets

A subset X in the Euclidean plane is called a k-distance set if there are exactly k distances between two distinct points in X. We denote the largest possible cardinality of k  -distance sets by g(k)g(k). Erdős and Fishburn proved that g(5)=12g(5)=12 and also conjectured that 12-point five-distance sets are unique up to similar transformations. We classify 8-point four-distance sets and prove the uniqueness of the 12-point five-distance sets given in their paper. We also introduce diameter graphs of planar sets and characterize these graphs.