Abstract
Bose et al. [1] asked whether for every simple arrangement \(\mathcal{A}\) of n lines in the plane there exists a simple n-gon P that induces \(\mathcal{A}\) by extending every edge of P into a line. We prove that such a polygon always exists and can be found in O(n logn) time. In fact, we show that every finite family of curves \(\mathcal{C}\) such that every two curves intersect at least once and finitely many times and no three curves intersect at a single point possesses the following Hamiltonian-type property: the union of the curves in \(\mathcal{C}\) contains a simple cycle that visits every curve in \(\mathcal{C}\) exactly once.
Research by Eyal Ackerman was supported by a fellowship from the Alexander von Humboldt Foundation. Research by Rom Pinchasi was supported by the Israeli Science Foundation (grant No. 938/06).
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Ackerman, E., Pinchasi, R., Scharf, L., Scherfenberg, M. (2009). On Inducing Polygons and Related Problems. In: Fiat, A., Sanders, P. (eds) Algorithms - ESA 2009. ESA 2009. Lecture Notes in Computer Science, vol 5757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04128-0_5
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DOI: https://doi.org/10.1007/978-3-642-04128-0_5
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