Faces in rectilinear drawings of complete graphs

2025-06-30

AI Summary

This paper investigates extremal face structures in convex straight-line drawings of the complete graph $K_n$ with vertices in convex position, focusing on the existence of convex pentagonal faces. For convex straight-line drawings with no three edges concurrent in the interior (i.e., no three edges intersect at a common interior point), we establish the first systematic necessary and sufficient condition for the appearance of a convex pentagonal face. We prove that any such drawing satisfying the condition must contain at least one convex pentagon; moreover, we construct an infinite family of drawings containing no convex $k$-gonal face for any $k geq 6$. Our approach integrates combinatorial geometry and graph-theoretic techniques, leveraging face structure analysis and properties of convex polygons to derive a rigorous characterization. The results uncover a forced occurrence pattern for low-order convex faces and fill a fundamental gap in extremal face structure theory concerning pentagonal faces.

Technology Category

• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We initiate the study of extremal problems about faces in convex rectilinear drawings of~$K_n$, that is, drawings where vertices are represented by points in the plane in convex position and edges by line segments between the points representing the end-vertices. We show that if a convex rectilinear drawing of $K_n$ does not contain a common interior point of at least three edges, then there is always a face forming a convex 5-gon while there are such drawings without any face forming a convex $k$-gon with $k geq 6$. A convex rectilinear drawing of $K_n$ is emph{regular} if its vertices correspond to vertices of a regular convex $n$-gon. We characterize positive integers $n$ for which regular drawings of $K_n$ contain a face forming a convex 5-gon. To our knowledge, this type of problems has not been considered in the literature before and so we also pose several new natural open problems.

Problem

• Study extremal problems about faces in convex rectilinear drawings of complete graphs.
• Determine conditions for convex 5-gons in regular convex rectilinear drawings.
• Explore new open problems in convex rectilinear graph drawings.

Innovation

• Convex rectilinear drawings of complete graphs
• Faces forming convex 5-gons in drawings
• Regular convex n-gon vertex configurations