Cubic plane graphs on a given point set
Jens M. Schmidt, Pável Valtr · 2012
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 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; the algorithm is constructive and runs in time O(n3). We also show which graph structure can be expected when there is a cubic plane graph on P; e.g., if P admits a 2-connected cubic plane graph, we show that P admits also a 2-connected cubic plane graph that contains the boundary cycle of P. The algorithm extends to checking P on admitting a 2-connected cubic plane graph.