Question: Problem 3 (5+ 5 = 10) HOPPING FROM ONE MINIMUM SPANNING TREE TO ANOTHER Let G be a weighted connected undirected graph, whose edge weights

Problem 3 (5+ 5 = 10) HOPPING FROM ONE MINIMUM SPANNING TREE TO ANOTHER Let G be a weighted connected undirected graph, whose edge weights are not necessarily distinct. (a) Suppose 17" and 7" are distinct minimum spanning trees for graph GG. Let ' be a lightest edge (smallest weight edge) among all edges that are in 7" and but not in 7. Show that there exists an edge T such that w(e) = w(e') and the tree obtained by replacing edge ' in 7" by is also a minimum spanning tree. (b) Using part (a), prove that if T and 7" are two minimum spanning trees of GG, then there exists a sequence (Ty, ..., 1), k > 0, such that: (i) 7; is a minimum spanning tree of G, 0

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