Chapter

Algorithms - ESA 2009

Volume 5757 of the series Lecture Notes in Computer Science pp 47-58

On Inducing Polygons and Related Problems

  • Eyal AckermanAffiliated withLancaster UniversityInstitute of Computer Science, Freie Universität Berlin
  • , Rom PinchasiAffiliated withLancaster UniversityMathematics Department, Technion—Israel Institute of Technology
  • , Ludmila ScharfAffiliated withLancaster UniversityInstitute of Computer Science, Freie Universität Berlin
  • , Marc ScherfenbergAffiliated withLancaster UniversityInstitute of Computer Science, Freie Universität Berlin

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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.