On Inducing Polygons and Related Problems

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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.