On Inducing Polygons and Related Problems

Bose et al. [1] asked whether for every simple arrangement A of n lines in the plane there exists a simple n-gon P that induces 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 log n) time. In fact, we show that every finite family of curves 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 C contains a simple cycle that visits every curve in C exactly once.