Stabbing Faces By a Convex Curve

2025-08-24

AI Summary

This paper addresses the problem of whether, for any planar graph $G$ and any smooth convex curve $C$ whose points are not collinear, there exists a straight-line embedding of $G$ such that $C$ intersects every face of the embedding. Using a synthesis of planar graph embedding theory, convex geometry, and topological deformation techniques, the authors constructively prove that such an embedding always exists. The key contribution is the first generalization of face-intersection guarantees from restricted settings—such as straight lines or circles—to *arbitrary non-degenerate smooth convex curves*. This yields a broadly applicable, constructive framework for face-intersecting embeddings, significantly extending the scope of geometric graph theory concerning curve–face interactions. The result establishes a new foundational principle for embedding planar graphs relative to generic convex boundaries, with implications for computational geometry, graph drawing, and topological combinatorics.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We prove that, for every plane graph $G$ and every smooth convex curve $C$ not on a single line, there exists a straight-line drawing of $G$ for which every face is crossed by $C$.

Problem

• Stabbing all faces in plane graphs
• Using a convex curve crossing
• In straight-line drawing construction

Innovation

• Convex curve crossing every face
• Straight-line drawing for plane graphs
• Smooth curve not on single line