On triconnected and cubic plane graphs on given point sets

Abstract

Let S be a set of n ⩾ 4 points in general position in the plane, and let h < n be the number of extreme points of S. We show how to construct a 3-connected plane graph with vertex set S, having max { ⌈ 3 n / 2 ⌉ , n + h − 1 } edges, and we prove that there is no 3-connected plane graph on top of S with a smaller number of edges. In particular, this implies that S admits a 3-connected cubic plane graph if and only if n ⩾ 4 is even and h ⩽ n / 2 + 1 . The same bounds also hold when 3-edge-connectivity is considered. We also give a partial characterization of the point sets in the plane that can be the vertex set of a cubic plane graph.