Harmonic functions on planar and almost planar graphs and ...

568 I. Benjamini, O. Schramm1.4. Definition A metric space X is said to have the C (R, r) property if for anytwo positive numbers R, r there is some finite C = C (R, r) such that any ball ofradius R in X can be covered by C balls of radius r.For example, a connected graph with its natural metric has the C (R, r) propertyiff there is a global bound on the degrees of its vertices.1.5. Definition A metric space X satisfying the C (R, r) property is almost planar,if there is a bounded valence planar graph G =(V,E)and a quasimonomorphismf : X → V , where V is equipped with the distance metric in G. A graph G ◦ =(V ◦ , E ◦ ) is almost planar, ifV ◦ with the distance metric of G ◦ is almost planar.We wish to stress that almost planarity is a much weaker property than beingroughly isometric to a planar graph. For example, take two copies of Z 2 andidentify them along the x-axis {(n, 0)}. The resulting graph is easily seen to bealmost planar, but is not roughly isometric to a planar graph.As explained below, if G 1 is a planar graph, and G 2 is finite, then anysubgraph of G 1 × G 2 is almost planar. This can be used to construct manyexamples of almost planar graphs.The following theorem gives an equivalent definition for almost planarity.1.6. Theorem Let X be a metric space that satisfies the C (R, r) property. ThenX is almost planar iff there exists a topologically planar, complete, Riemanniansurface S with bounded curvature, and a quasimonomorphism f : X → S.An easy, but useful, observation is:1.7. Fact In the category of metric spaces satisfying the C (R, r) property, a roughisometry is a quasimonomorphism, and the composition of quasimonomorphismsis a quasimonomorphism. Hence, for such spaces, almost planarity is invariantunder rough isometries.⊓⊔This, in particular, shows that when G 1 is an almost planar graph and G 2 isa finite graph, then any subgraph of G 1 × G 2 is almost planar.Regarding almost planar graphs, we have the following geometric criterion.1.8. Theorem Let G be a connected, bounded degree graph. Then G is almostplanar iff there is some finite κ such that G can be drawn in the plane in such away that every edge has at most κ crossings.We shall now clarify the statement of the theorem, by defining what a crossingis. Given a graph G, welet|G|denote the metric space constructed as follows.Start with V , and for every edge e ∈ E, with vertices v, u, say, glue the endpointsof an isometric copy I e of the interval [0, 1] to the two vertices v, u. Let |G| bethe union ∪ e I e ∪ V modulo the identifictions, with the path metric. A drawing of