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′.