Fast Algorithms for Diameter-Optimally Augmenting Paths and Trees
  We consider the problem of augmenting an n-vertex graph embedded in a metric space, by inserting one additional edge in order to minimize the diameter of the resulting graph. We present exact algorithms for the cases when (i) the input graph is a path, running in O(n \log^3 n) time, and (ii) the input graph is a tree, running in O(n^2 \log n) time. We also present an algorithm that computes a (1+\eps)-approximation in O(n + 1/\eps^3) time, for paths in R^d, where d is a constant. 
Ulrike Gro{\ss}e, Joachim Gudmundsson, Christian Knauer, Michiel Smid   and Fabian Stehn
cs.CG