  Given a set S of n points, a weight function w to associate a non-negative
weight to each point in S, a positive integer k \ge 1, and a real number
\epsilon > 0, we present algorithms for computing a spanner network G(S, E) for
the metric space (S, d_w) induced by the weighted points in S. The weighted
distance function d_w on the set S of points is defined as follows: for any p,
q \in S, d_w(p, q) is equal to w(p) + d_\pi(p, q) + w(q) if p \ne q, otherwise,
d_w(p, q) is 0. Here, d_\pi(p, q) is the Euclidean distance between p and q if
points in S are in \mathbb{R}^d, otherwise, it is the geodesic (Euclidean)
distance between p and q. The following are our results: (1) When the weighted
points in S are located in \mathbb{R}^d, we compute a k-vertex fault-tolerant
(4+\epsilon)-spanner network of size O(k n). (2) When the weighted points in S
are located in the relative interior of the free space of a polygonal domain
\cal P, we detail an algorithm to compute a k-vertex fault-tolerant
(4+\epsilon)-spanner network with O(\frac{kn\sqrt{h+1}}{\epsilon^2} \lg{n})
edges. Here, h is the number of simple polygonal holes in \cal P. (3) When the
weighted points in S are located on a polyhedral terrain \cal T, we propose an
algorithm to compute a k-vertex fault-tolerant (4+\epsilon)-spanner network,
and the number of edges in this network is O(\frac{kn}{\epsilon^2} \lg{n}).
