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.