Rectangular Duals on the Cylinder and the Torus
Abstract
A rectangular dual of a plane graph is a contact representation of 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.
Cite
@article{arxiv.2506.07170,
title = {Rectangular Duals on the Cylinder and the Torus},
author = {Therese Biedl and Philipp Kindermann and Jonathan Klawitter},
journal= {arXiv preprint arXiv:2506.07170},
year = {2025}
}