On embedding triconnected cubic graphs on point sets

We show that for a given set S of 2n points in general position in the plane, there is a cubic 3-connected crossing-free geometric graph on the vertex set S if and only if the interior of the convex hull of S contains at least n − 1 points of S. We also give similar results for sets of odd cardinality. Analogous partial results are given if the 3-connected crossing-free geometric graph is not required to be cubic.