Sparse geometric graphs with small dilation

Given a set S of n points in the plane, and an integer k such that 0 ??? k < n, we show that a geometric graph with vertex set S, at most n – 1 + k edges, and dilation O(n / (k + 1)) can be computed in time O(n log n). We also construct n–point sets for which any geometric graph with n – 1 + k edges has dilation ??(n / (k + 1)); a slightly weaker statement holds if the points of S are required to be in convex position