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Covering islands in plane point sets. (English) Zbl 1374.52021
Márquez, Alberto (ed.) et al., Computational geometry. XIV Spanish meeting on computational geometry, EGC 2011, dedicated to Ferran Hurtado on the occasion of his 60th birthday, Alcalá de Henares, Spain, June 27–30, 2011. Revised selected papers. Berlin: Springer (ISBN 978-3-642-34190-8/pbk). Lecture Notes in Computer Science 7579, 220-225 (2012).
Summary: Let \(S\) be a set of \(n\) points in general position in the plane. A \(k\)-island \(I\) of \(S\) is a subset of \(k\) points of \(S\) such that \(\mathrm{Conv}(I) \cap S = I\). We show that, for an arbitrary but fixed number \(k \geq 2\), the minimum number of \(k\)-islands among all sets \(S\) of \(n\) points is \(\Theta (n^{2})\). The following related counting problem is also studied: For \(l < k\), an \(l\)-island covers a \(k\)-island if it is contained in the \(k\)-island. Let \(C_{k,l}(S)\) be the minimum number of \(l\)-islands needed to cover all the \(k\)-islands of \(S\) and let \(C _{k,l }(n)\) be the minimum of \(C_{k,l }(S)\) among all sets \(S\) of \(n\) points. We show asymptotic bounds for \(C _{k,l }(n)\).
For the entire collection see [Zbl 1253.68016].
MSC:
52C10 Erdős problems and related topics of discrete geometry
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