Handbook of Graph Theory, 2nd Edition

Chapter 7

Topological Graph Theory

Section 7.1 Graphs on Surfaces

Tomaž Pisanski

University of Ljubljana, Slovenia

Primož Potocnik

University of Ljubljana, Slovenia

Introduction

The need to imbed (draw) finite graphs on surfaces arises in various aspects of mathematics and science. Often the simplest surface in which such a graph can be imbedded is sought. Some generalizations of surfaces are briefly considered.

7.1.1 Surfaces

2-Manifolds and 2-Pseudomanifolds

Definitions

D1: The open unit disk, the closed unit disk, and the unit half-disk are the respective subsets

$\left\{\left(x,y\right)|x^2+y^2<1\right\},\left\{\left(x,y\right)|x^2+y^2\le 1\right\},\text{and}\left\{\left(x,y\right)|x\ge 0,x^2+y^2<1\right\}$

of the Euclidean plane, together with the inherited Euclidean ...