1. (20pts) Let G (V, E) be an edge-weighted connected graph, and T- (V,E') be its minimum spanning tree, and e be any edge in T a) consider the graph T'-(V, E,-(e)). How many connected component(s) does T' have? Prove your answer from the definition of a connected component. b) Let C' be any connected component in T', and let E" S E be the set of edges connecting C' and V - C. That is, E',-{(x, y) : (x, y) E E, x E C'.y E V-C'}, edge (x, y) in G=(V, E) such that the first node x is in C, and the second node y is not. This edge set E" ie, including every is said to be the cut (C', V- C') of G playing an important role in computer science. Prove that the given edge e has the minimum weight among all in the cut (C',V - C') 1. (20pts) Let G (V, E) be an edge-weighted connected graph, and T- (V,E') be its minimum spanning tree, and e be any edge in T a) consider the graph T'-(V, E,-(e)). How many connected component(s) does T' have? Prove your answer from the definition of a connected component. b) Let C' be any connected component in T', and let E" S E be the set of edges connecting C' and V - C. That is, E',-{(x, y) : (x, y) E E, x E C'.y E V-C'}, edge (x, y) in G=(V, E) such that the first node x is in C, and the second node y is not. This edge set E" ie, including every is said to be the cut (C', V- C') of G playing an important role in computer science. Prove that the given edge e has the minimum weight among all in the cut (C',V - C')