  In this paper, we study weakly dynamic undirected graphs, that can be used to
represent some logistic networks. The goal is to deliver all the delivery
points in the network. The network exists in a mostly stable environment,
except for a few edges known to be non-stable. The weight of each of these
non-stable edges may change at any time (bascule or lift bridge, elevator,
traffic congestion...). All other edges have stable weights that never change.
This problem can be now considered as a Minimum Spanning Tree (MST) problem on
a dynamic graph. We propose an efficient polynomial algorithm that computes in
advance alternative MSTs for all possible configurations. No additional
computation is then needed after any change in the problem because the MSTs are
already known in all cases. We use these results to compute critical values for
the non-stable weights and to pre-compute best paths. When the non-stable
weights change, the appropriate MST may then directly and immediately be used
without any recomputation.
