On saturated triangulation-free convex geometric graphs

2025-08-18

AI Summary

This paper investigates saturated graphs without triangulations in convex geometric graphs—graphs that contain no triangulation but become triangulated upon adding any missing edge. It introduces the notion of saturation to this geometric graph theory problem for the first time, integrating combinatorial geometry and extremal graph theory. Through structural analysis of diagonal configurations in convex $n$-gons and constructive proofs, the authors characterize the possible structures and edge counts of such saturated graphs. Key contributions are: (1) construction of saturated graphs with only $O(n log n)$ edges, breaking the conventional paradigm of dense constructions; (2) proof that the number of edges in saturated graphs attains all integer values in the continuous range from $O(n log n)$ to $inom{n}{2} - 1$; and (3) complete classification of all saturated graphs achieving the maximum possible edge count minus one, i.e., $inom{n}{2} - 1$. These results establish a full spectrum of saturated graphs—from sparse to nearly complete—revealing rich structural transitions.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Social Networks and Social Media: Social media analysis through the lenses of networks

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.

Problem

• Characterizing saturated triangulation-free convex geometric graphs
• Studying minimal edge addition that creates forbidden triangulations
• Determining possible edge counts for saturated graphs

Innovation

• Saturated triangulation-free convex geometric graphs
• Constructed saturated graphs with O(n log n) edges
• Characterized saturated graphs with n choose 2 minus n-1 edges