Gruber P. Convex and Discrete Geometry

504 Geometry of Numbers
Then H is a planar representation of G ′ , i.e. it has no edge crossings.
If J is a closed Jordan curve in C, we denote by interior J the bounded component
of C\J. LetC∞ be the cycle of the exterior country of G ′ .
To see (3), we first extend f to a continuous map of the point set of G ′ , i.e.
the union of all vertices and edges of G ′ , onto the point set of H such that f
is one-to-one on each edge of G ′ . Next, we extend f step by step to a covering
map of C onto C: Noting (iii), we may extend f by the Jordan-Schönflies theorem
for each country cycle C �= C∞ of G ′ to interior C such that the extended f
maps C ∪ interior C homeomorphically onto the compact set f (C) ∪ interior f (C).
The extended f then maps C∞∪ interior C∞ continuously onto the compact set
f (C∞) ∪ f (interior C∞). Letx ∈ interior C∞. By distinguishing the cases where
x is an interior point of a bounded country, a relatively interior point of an edge,
or a vertex of G ′ and noting (iv) and (ii), we see that there are neighbourhoods U
of x and W of f (x) such that f maps U homeomorphically onto W . As a consequence,
we see that f (interior C∞) is an open set in E 2 . Since a boundary point of
f (interior C∞) must be a limit point of points of the form f (xn), n = 1, 2,..., where
xn ∈ interior C∞,itmustbeoftheform f (x) where x ∈ C∞∪interior C∞. (Consider
a convergent subsequence of the sequence (xn).) Since x ∈ interior C∞ is excluded,
x ∈ C∞. Hence bd f (interior C∞) ⊆ f (C∞). Thus, noting that f (interior C∞)
is open, connected and bounded, f (interior C∞) = interior f (C∞) follows. By a
version of the Jordan-Schönflies theorem, f finally can be extended to a continuous
map f : C → C such that f is a homeomorphism on C\interior C∞. Let
x ∈ C\interior C∞. As before, we see that there are neighbourhoods U of x and W
of f (x) such that f maps U homeomorphically onto W . The extended f satisfies the
required assumptions of (2) and thus is a homeomorphism of C onto C. In particular,
H is a planar representation of G ′ , concluding the proof of (3).
In the fourth step we show that the conditions (1)(i,ii) yield a weak primal-dual
circle packing of G:
(4) Let ρ = � ρv : v ∈ V(G ′ ) � satisfy (1)(i,ii). Then there is a weak primal-dual
circle packing of G with radii ρ = (ρv) and with the same local clockwise
orientation as in G and G ∗ .
We shall construct an isomorphic planar image of G ′ such that the image of an edge
uv is a line segment. For each vertex u ∈ V(G ′ ) we shall determine a point ū ∈ C
such that the discs with centres ū and radii ρu form the desired weak primal-dual
circle packing. We start with an edge uv ∈ E(G ′ ) and draw a corresponding line
segment ū ¯v in C of length � ρ2 u + ρ2 �1/2. v Then the position of each neighbour of ū is
uniquely determined. Now, given an arbitrary vertex w ∈ V(G ′ ), consider a path in
G ′ connecting u and w and use it to construct ¯w. For the proof that ¯w is independent
of the path chosen, it is sufficient to show that, for each simply closed path in G ′ , our
construction leads to a closed path in C. This can easily be shown by induction on
the number of country 4-cycles in the simply closed path. The image of G ′ in C thus
obtained satisfies the properties specified in (3). Hence it has no crossings and thus
is a planar graph. For each of its vertices ū the disc Du with centre ū and radius ρu
is contained in the convex polygon consisting of the convex kites which correspond