The optimality of the running time follows from Chen et al. If k is a constant, then the best result is that of Theorem 2. It gives a Euclidean k fault tolerant t spanner, in which the degree of each vertex is bounded by a constant. Chen et al. Therefore, the result of Theorem 2 is optimal for constant values of k.
Some interesting problems remain to be solved.
D. Z. Chen, G. Das, and M. Smid. Lower bounds for computing geometric spanners and approximate shortest paths . This follows from the simple fact that in such a spanner the closest pair must be connected by an edge. Chew shows that the L 1 Delaunay triangulation of S is a Euclidean p 10 spanner. Lower bounds for computing geometric spanners and approximate shortest paths .