Gruber P. Convex and Discrete Geometry

34 Koebe’s Representation Theorem for Planar Graphs 505
ū
Du
Dv
Fig. 34.4. Kite
to the country 4-cycles in G ′ with vertex u (Fig. 34.4). Since this holds for any vertex,
the disc Du touches only those discs Dv or their boundaries intersect orthogonally,
for which ¯v is a vertex of a kite with vertex ū. Hence the discs Du : u ∈ V(G ′ ) form
a weak primal-dual circle packing of G, concluding the proof of (4).
In the fifth step we refine (4):
(5) Let ρ = � ρv : v ∈ V(G ′ ) � satisfy (1)(i,ii). Then there exists a primal-dual
circle packing of G.
Assume first that the outer cycle of G is a 3-cycle v1v2v3. By (4) there is a weak
primal-dual circle packing of G. By applying a suitable Möbius transformation and
changing notation, if necessary, we obtain a weak primal-dual circle packing of G
and such that ¯v1, ¯v2, ¯v3 are the vertices of a regular triangle with centroid 0. Since
the discs Dv1 , Dv2 , Dv3 with centres ¯v1, ¯v2, ¯v3 touch pairwise, they all have the
same radius. Hence there is a circular disc D∞ with centre ∞ such that its boundary
circle intersects orthogonally the boundary circles of the discs Dv1 , Dv2 , Dv3
at the points where these discs touch pairwise. This yields the desired primal-dual
circle packing of G (Fig. 34.5). If, second, the outer cycle of G has length greater
than 3, then either G or G∗ has a country 3-cycle by the Corollary 15.1 of the Euler
polytope formula. Interchanging the roles of G and G∗ , if necessary, and applying a
suitable Möbius transformation, if necessary, we arrive at the same situation as in the
previous paragraph. Thus there exists a corresponding primal-dual circle packing.
The inverse of the Möbius transformation then yields the desired primal-dual circle
packing of G.
In the sixth step we prepare the way for the seventh step:
(6) Let S � V(G ∧ ) with #S ≥ 5. Then 2#S − #E � G ∧ (S) � ≥ 5.
Here G ∧ (S) is the subgraph of G ∧ with vertex set S whose edges are precisely
the edges uv of G ∧ with u,v ∈ S. To see (6), note first that the following
holds:
¯v