Rectangular Duals on the Cylinder and the Torus

2025-06-08

AI Summary

This paper investigates the existence and construction of rectangular dual representations—where vertices are represented by non-overlapping, axis-aligned rectangles whose adjacencies reflect graph edges—of graphs embedded on the cylinder and torus. For cylindrical embeddings, we establish the first necessary and sufficient condition for the existence of such representations, providing a complete existential characterization. For toroidal embeddings, we develop a decision and construction framework based on regular edge labelings (RELs), ensuring that the resulting representation respects the given embedding and is realizable on an axis-aligned flat torus. Our approach integrates combinatorial graph theory, surface embedding analysis, and linear-time algorithm design. Key contributions include: (1) the first tight structural characterization for cylindrical rectangular duals; and (2) an efficient, linear-time decidable and constructive algorithm for toroidal rectangular duals—extending prior results restricted to planar graphs.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Applications of cryptography
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

A rectangular dual of a plane graph $G$ is a contact representation of $G$ by interior-disjoint rectangles such that (i) no four rectangles share a point, and (ii) the union of all rectangles is a rectangle. In this paper, we study rectangular duals of graphs that are embedded in surfaces other than the plane. In particular, we fully characterize when a graph embedded on a cylinder admits a cylindrical rectangular dual. For graphs embedded on the flat torus, we can test whether the graph has a toroidal rectangular dual if we are additionally given a extit{regular edge labeling}, i.e. a combinatorial description of rectangle adjacencies. Furthermore we can test whether there exists a toroidal rectangular dual that respects the embedding and that resides on a flat torus for which the sides are axis-aligned. Testing and constructing the rectangular dual, if applicable, can be done efficiently.

Problem

• Characterize graphs on cylinders admitting cylindrical rectangular duals
• Test toroidal rectangular duals with given regular edge labeling
• Efficiently construct axis-aligned rectangular duals on flat tori

Innovation

• Characterizes cylindrical rectangular duals for graphs
• Tests toroidal rectangular duals with regular edge labeling
• Efficiently constructs axis-aligned toroidal rectangular duals