Abstract
Let G be a planar graph such that the volume function of G satisfies V(2n)< CV(n) for some constant C > 0. Then for every vertex v of G and integer n, there is a domain Ωsuch that B(v,n) \subset Ω, Ω\subset B(v, 6n) and the size of the boundary of Ωis at most order n.
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