On saturated triangulation-free convex geometric graphs

TL;DR

This paper explores the properties of convex geometric graphs that are saturated with respect to triangulation, revealing the existence of such graphs with surprisingly few edges and providing a complete characterization for certain edge counts.

Contribution

It introduces the saturation problem for triangulation-free convex geometric graphs, demonstrating the existence of sparse saturated graphs and characterizing those with nearly maximum edges.

Findings

Existence of saturated graphs with O(n log n) edges.

Construction of saturated graphs with any number of edges between g(n) and the maximum.

Complete characterization of saturated graphs with one less than maximum edges.

Abstract

A convex geometric graph is a graph whose vertices are the corners of a convex polygon P in the plane and whose edges are boundary edges and diagonals of the polygon. It is called triangulation-free if its non-boundary edges do not contain the set of diagonals of some triangulation of P. Aichholzer et al. (2010) showed that the maximum number of edges in a triangulation-free convex geometric graph on n vertices is ${{n}\choose{2}}-(n-2)$, and subsequently, Keller and Stein (2020) and (independently) Ali et al. (2022) characterized the triangulation-free graphs with this maximum number of edges. We initiate the study of the saturation version of the problem, namely, characterizing the triangulation-free convex geometric graphs which are not of the maximum possible size, but yet the addition of any edge to them results in containing a triangulation. We show that, surprisingly, there exist saturated graphs with only g(n) = O(n log n) edges. Furthermore, we prove that for any $n > n_0$ and any $g(n)\leq t \leq {{n}\choose{2}}-(n-2)$, there exists a saturated graph with n vertices and t edges. In addition, we obtain a complete characterization of all saturated graphs whose number of edges is ${{n}\choose{2}}-(n-1)$, which is 1 less than the maximum.