Universal point sets for planar three-trees

For every n∈Nn∈N, we present a set SnSn of O(n3/2log⁡n)O(n3/2log⁡n) points in the plane such that every planar 3-tree with n   vertices has a straight-line embedding in the plane in which the vertices are mapped to a subset of SnSn. This is the first subquadratic upper bound on the cardinality of universal point sets for planar 3-trees, as well as for the class of 2-trees and serial parallel graphs.