Gruber P. Convex and Discrete Geometry

500 Geometry of Numbers
Verdière [214,215]. An extension of Koebe’s theorem to a representation of G and its
dual G ∗ by two related packings of discs is due to Brightwell and Scheinerman [168].
These results have a series of important consequences: Miller and Thurston
(unpublished) showed that Koebe’s theorem yields the basic theorem of Lipton and
Tarjan [661] on separation of graphs, see Miller, Teng, Thurston and Vavaies [724]
and Pach and Agarwal [783]. Koebe’s theorem and the corresponding algorithms
of Thurston [999], Rodin and Sullivan [844] and Mohar [747] yield constructive
approximations to the analytic functions as in the Riemann mapping theorem. The
extension of Brightwell and Scheinerman proves a conjecture of Tutte [1002] on
simultaneous straight line representation of G and its dual G ∗ . For us, it is important
that the result of Brightwell and Scheinerman readily yields a refined version of the
representation theorem 15.6 of Steinitz for convex polytopes in E 3 .
In this section we first present the theorem of Brightwell and Scheinerman, then
discuss the algorithm of Thurston which yields a construction of the circle packing
corresponding to a graph with triangular countries that is, for Koebe’s theorem, and
outline how this can be used to obtain approximations for the Riemann mapping
theorem.
For more information and references to the literature, see the books of Pach
and Agarwal [783] on combinatorial geometry, of Mohar and Thomassen [748] and
Felsner [332] on graphs and of Stephenson [967] on packings of circular discs in the
context of discrete analytic functions. See also Sachs [872] and Stephenson [966].
34.1 The Extension of Koebe’s Theorem by Brightwell and Scheinerman
After introducing needed terminology on planar graphs, we present a proof of the
theorem of Brightwell and Scheinerman.
Graph Terminology
Recall the definitions and notation in Sect. 15.1. Let G be a 3-connected planar graph
in C ∪{∞}. We assume that ∞ is not a vertex of G. Since G is 3-connected, any
two countries of G are either disjoint, or have one common vertex, or one common
edge. The exterior country of G is the country containing the point ∞ in its interior.
Up to isomorphisms the dual graph G ∗ of G is defined as follows: In each country
of G, including the exterior country, choose a point. These points are the vertices
of G ∗ . Distinct vertices of G ∗ are connected by an edge in G ∗ if the corresponding
countries of G have an edge of G in common. Clearly, G ∗ can be drawn in C ∪{∞}.
It can be shown that G ∗ is also 3-connected and planar. Next, the vertex-country
incidence graph G ∧ of G will be defined: Its vertices are the vertices of G and G ∗ .
An edge of G ∧ connects a vertex v of G with a vertex w of G ∗ if v is a vertex of the
country of G corresponding to w. There are no other edges. By applying a suitable
Möbius transformation, if necessary, we may assume that ∞ is the vertex of G ∗
which corresponds to the exterior country of G. LetG ′ denote the graph obtained
from G ∧ by deleting the vertex ∞ and all edges incident with it. See Bollobás [142]
and Mohar and Thomassen [748] (Fig. 34.1).