Cubic plane graphs on a given point set

Let P be a set of n>=4 points in the plane that is in general position and such that n is even. We investigate the problem whether there is a (0-, 1- or 2-connected) cubic plane straight-line graph on P. No polynomial-time algorithm is known for this problem. Based on a reduction to the existence of certain diagonals of the boundary cycle of the convex hull of P, we give the first polynomial-time algorithm that checks for 2-connected cubic plane graphs; the algorithm is constructive and runs in time O(n^3). We also show which graph structure can be expected when there is a cubic plane graph on P; e.@?g., a cubic plane graph on P implies a connected cubic plane graph on P, and a 2-connected cubic plane graph on P implies a 2-connected cubic plane graph on P that contains the boundary cycle of P. We extend the above algorithm to check for arbitrary cubic plane graphs in time O(n^3).