Consider the Minimum Spanning Tree Problem on an undirected graph G = (V, E), with a cost...

Consider the Minimum Spanning Tree Problem on an undirected graph G = (V, E), with a cost ce ≥ 0 on each edge, where the costs may not all be different. If the costs are not all distinct, there can in general be many distinct minimum-cost solutions. Suppose we are given a spanning tree  with the guarantee that for every e e T, e belongs to some minimum-cost spanning tree in G. Can we conclude that T itself must be a minimum-cost spanning tree in G? Give a proof or a counterexample with explanation.