Let C be a polygonal cycle on n vertices in the plane. A randomized algorithm is presented which computes in O(n log3 n) expected time, the edge of C whose removal results in a polygonal path of smallest possible dilation. It is also shown that the edge whose removal gives a polygonal path of largest possible dilation can be computed in O(n log n) time. If C is a convex polygon, the running time for the latter problem becomes O(n). Finally, it is shown that for each edge e of C, a (1 - ε)-approximation to the dilation of the path C \ {e} can be computed in O(n log n) total time.