Visibility Representations of Toroidal and Klein-bottle Graphs

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.