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.