Triangulability of Convex Graphs and Convex Skewness

TL;DR

This paper investigates conditions under which the removal of certain subgraphs from convex complete graphs still allows for triangulation, and applies these findings to determine convex skewness of such graphs.

Contribution

It provides necessary and sufficient conditions for triangulability after removing subgraphs with up to n-1 edges and explores packing strategies for larger subgraphs.

Findings

Characterizes when K_n - F admits a triangulation for |E(F)| q n-1.

Identifies conditions for packing F in K_n for |E(F)|  n.

Determines convex skewness of graphs of the form K_n - F.

Abstract

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.