Visibility Representations of Toroidal and Klein-bottle Graphs

TL;DR

This paper proves that graphs embedded on a torus or Klein bottle can be represented with visibility representations on rectangular flat surfaces, answering a longstanding open question.

Contribution

It establishes that all toroidal and Klein-bottle embedded graphs have rectangular visibility representations, extending previous results to rectangular flat surfaces.

Findings

Toroidal graphs have rectangular visibility representations.

Klein-bottle graphs also have rectangular visibility representations.

The results answer open questions in topological graph theory.

Abstract

In this paper, we study visibility representations of graphs that are embedded on a torus or a Klein bottle. Mohar and Rosenstiehl showed that any toroidal graph has a visibility representation on a flat torus bounded by a parallelogram, but left open the question whether one can assume a rectangular flat torus, i.e., a flat torus bounded by a rectangle. Independently the same question was asked by Tamassia and Tollis. We answer this question in the positive. With the same technique, we can also show that any graph embedded on a Klein bottle has a visibility representation on the rectangular flat Klein bottle.