Given a graph G and a minimum spanning tree T, suppose that we decrease the weight of one of the edges in T. Show that T is still a minimum spanning tree for G. More formally, let T be a minimum spanning tree for G with edge weights given by weight function w. Choose one edge (x, y) ¬ T and a positive number k, and define the weight function w′ by




Show that T is a minimum spanning tree for G with edge weights given by w′.

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if (u, v) # (x, y). w(u, v) w(x, y) - k if (u, v) = (x, y). w'(u, v) =