On point-sets that support planar graphs

A universal point-set supports a crossing-free drawing of any planar graph. For a planar graph with n vertices, if bends on edges of the drawing are permitted, universal point-sets of size n are known, but only if the bend points are in arbitrary positions. If the locations of the bend points must also be specified as part of the point-set, we prove that any planar graph with n vertices can be drawn on a universal set S of O(n(2) / logn) points with at most one bend per edge and with the vertices and the bend points in S. If two bends per edge are allowed, we show that 0(n logn) points are sufficient, and if three bends per edge are allowed, 0(n) points are sufficient. When no bends on edges are permitted, no universal point-set of size o(n(2)) is known for the class of planar graphs. We show that a set of n points in balanced biconvex position supports the class of maximum-degree-3 series-parallel lattices. (C) 2012 Elsevier B.V. All rights reserved.