Triangulability of Convex Graphs and Convex Skewness

Motivated by a result of [1] which states that if F is a subgraph of a convex complete graph K_n and F contains no boundary edge of K_n and |E(F)| \leq n-3, then K_n - F admits a triangulation, we determine necessary and sufficient conditions on F with |E(F)| \leq n-1 for which the conclusion remains true. For |E(F)| \geq n, we investigate the possibility of packing F in K_n such that K_n -F admits a triangulation for certain families of graphs F. These results are then applied to determine the convex skewness of the convex graphs of the form K_n - F.