Let s and t be two vertices in a undirected graph G + (V, E) having distinct…

Answer: a path from s to t in the minimum weighted spanning tree (MST).

Key property: In an MST, the path between any two vertices minimizes the maximum edge weight along the path (the bottleneck). Therefore the MST path between s and t minimizes congestion.

• Let T be a minimum spanning tree and let P_T be the unique path from s to t in T. Let e be the heaviest edge on P_T, with weight w*.

• Assume for contradiction there is another s–t path P' whose every edge has weight strictly less than w*.

• Removing e from T splits T into two components X and Y with s in X and t in Y. Path P' must contain at least one edge f that goes between X and Y.

• By assumption the weight of f is less than w*. Adding f to T creates a cycle that contains e, and e is the heaviest edge on that cycle. Replacing e by f yields a spanning tree with smaller total weight, contradicting the minimality of T.

• Thus no s–t path has all edges lighter than w*, so the MST path P_T attains the smallest possible maximum-edge weight among all s–t paths. Hence it minimizes congestion.

This proves that the path between s and t inside a minimum spanning tree is always a path of minimum congestion.