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
w'(u, v) = w(u, v), if (u, v) != (x, y),
w(u, v) − k, if (u, v) = (x, y).
Show that T is a minimum spanning tree for G with edge weights given by w'