Combinatorial complexity of signed discs
Let C+ and C− be two collections of topological discs of arbitrary radii. The collection of discs is ‘topological’ in the sense that their boundaries are Jordan curves and each pair of Jordan curves intersect at most twice. We prove that the region ∪C+−∪C− has combinatorial complexity at most 10n-30 where p=¦C+¦, q=¦C−¦ and n=p+q≥5. Moreover, this bound is achievable. We also show bounds that are stated as functions of p and q. These are less precise.
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