Let G be an undirected graph, where additionally each edge e E has a positive real valued cost ce associated with it. A nice tree T of G is minimal if the sum of the costs ce of edges belonging to T is minimal among the set of such sums over the set of all nice trees of G. The bottleneck of a path p in G is the maximum cost maxecp ce of one of its edges. An edge {v, w} E satisfies the bottleneck property if it is a minimum-bottleneck path between vand w. Prove that the following are equivalent: Ce every edge of a nice tree T of a graph G with all edge costs being distinct satisfies the bottleneck property T is a minimal nice tree.