Combinatorial complexity of signed discs

Let $CC^+$ 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.Technical report LCSR-TR-19