Rectangular Duals on the Cylinder and the Torus

TL;DR

This paper extends the concept of rectangular duals from planar graphs to graphs embedded on cylinders and tori, providing characterizations, testing algorithms, and construction methods for these non-planar surfaces.

Contribution

It characterizes when graphs on a cylinder admit cylindrical rectangular duals and develops efficient algorithms for testing and constructing toroidal rectangular duals with given embeddings.

Findings

Characterization of cylindrical rectangular duals.

Efficient algorithms for testing toroidal rectangular duals.

Construction methods for duals respecting embeddings.

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 \textit{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.