AbstractLet $CC^+$ and $CC^-$ 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 $cupCC^+
-cupCC^-$ has combinatorial complexity at most $10n-30$ where
$p=|CC^+|$, $q=|CC^-|$ and $n=p+qge 5$. 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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