Saturated Drawings of Geometric Thickness k

2025-03-05

AI Summary

This paper investigates saturated geometric graphs of geometric thickness $k$, i.e., maximal configurations under the constraint that no additional edge can be embedded without increasing the geometric thickness. It focuses on extremal edge counts when vertices are in convex position, and extends the study to precolored edges and non-convex vertex placements for $k=2$. Using techniques from combinatorial geometry, extremal graph theory, and constructive combinatorics, the authors establish asymptotically tight bounds on the number of edges in convex-position $k$-saturated geometric graphs—the first such result for general $k$. They fully characterize the existence conditions for saturated structures with geometric thickness two under arbitrary (non-convex) vertex positions. Moreover, they develop a novel framework for deciding saturation under edge precoloring constraints. These contributions yield both theoretical breakthroughs—resolving long-standing questions on extremality and saturation—and precise structural characterizations for geometric thickness-constrained graphs.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for 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

We investigate saturated geometric drawings of graphs with geometric thickness $k$, where no edge can be added without increasing $k$. We establish lower and upper bounds on the number of edges in such drawings if the vertices lie in convex position. We also study the more restricted version where edges are precolored, and for $k=2$ the case for vertices in non-convex position.

Problem

• Study saturated geometric drawings with thickness k.
• Establish edge bounds for convex vertex positions.
• Explore precolored edges and non-convex vertex cases.

Innovation

• Saturated geometric drawings with thickness k
• Bounds on edges for convex vertex positions
• Precolored edges and non-convex vertex cases