Tesselation and visibility representations of maps on the torus
type: Item
instance of: scholarly article
title: Tesselation and visibility representations of maps on the torus
zbMATH Open document ID: 0901.52024
DOI: 10.1007/PL00009344
published in: Discrete \& Computational Geometry
publication date: 1998-11-22T00:00:00Z
review text: A well-known model of the torus is given by a parallelogram whose opposite sides are pairwise identified. An important concept related to drawing of graphs in the plane is the notion of upward drawings where each edge of a graph is oriented and drawn in \(R^2\) such that the ordinate monotonically increases when we traverse the edge according to its orientation. In view of the parallelogram mentioned above, the authors define two families of parallel lines and tessellate the torus, and then they associate to every tesselation a toroidal map with an upward drawing. On this basis they prove that a toroidal map admits a tessellation representation if and only if it is essentially 2-connected. Further on, several characterizations of essentially 2-connected maps on the torus are given, and it is shown that they can be obtained from two minimal maps by vertex splitting and creating digons. And also those graphs are characterized that can be embedded in the torus as an essentially 2-connected map.
Mathematics Subject Classification ID: 52C99
zbMATH DE Number: 1127609
zbMATH Keywords: tessellation
zbMATH Keywords: torus
zbMATH Keywords: Euler characteristic
zbMATH Keywords: dual map
zbMATH Keywords: upward drawing of a graph
zbMATH Keywords: 2-connected graphs
zbMATH Keywords: minimal map
author: Bojan Mohar
author: Pierre Rosenstiehl
reviewed by: Horst Martini
MaRDI profile type: Publication
full work available at URL: pl00009344
OpenAlex ID: W2046557343
description: scientific article; zbMATH DE number 1127609