Jeff Erickson

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  Untangling Planar Curves and Planar Graphs
  Jeff Erickson
  Any generic closed curve in the plane can be transformed into a simple closed curve by
  a finite sequence of local transformations called homotopy moves. We prove that simplifying a
  planar curve with n self-crossings requires Theta(n^{3/2}) homotopy moves in the worst case.
  The best bounds previously known were a 100-year-old O(n^2) upper bound due to Steinitz and
  the trivial Omega(n) lower bound. Our lower bound also implies that Omega(n^{3/2}) degree-1
  reductions, series-parallel reductions, and Delta-Y transformations are required to reduce any
  planar graph with treewidth Omega(sqrt{n}) to a single edge, matching known upper bounds for
  rectangular and cylindrical grid graphs. Finally, we prove that Omega(n^2) homotopy moves are
  required in the worst case to transform one non-contractible closed curve on the torus to another;
  this lower bound is tight if the curve is homotopic to an embedding.