Constrained tri-connected planar straight line graphs

It is known that for any set V of n ≥ 4 points in the plane, not in convex position, there is a 3-connected planar straight line graph G = (V, E) with at most 2n - 2 edges, and this bound is the best possible. We show that the upper bound |E| ≤ 2n continues to hold if G = (V, E) is constrained to contain a given graph G ₀ = (V, E ₀), which is either a 1-factor (i.e., disjoint line segments) or a 2-factor (i.e., a collection of simple polygons), but no edge in E₀ is a proper diagonal of the convex hull of V. Since there are 1- and 2-factors with n vertices for which any 3-connected augmentation has at least 2n - 2 edges, our bound is nearly tight in these cases. We also examine possible conditions under which this bound may be improved, such as when G₀ is a collection of interior-disjoint convex polygons in a triangular container.