I need a proof as a solution for both of these questions. (a) Suppose G = (V, E) is any undirected graph and (S, V-S) is a cut with zero cut edges in G. Provethat if we pick two arbitrary vertices u ES and v € V - S, and add a new edge (u, v), in the resultinggraph, there is no cycle that contains the edge (u, v). (b) Suppose G = (V, E) is an undirected graph with weight we on each edge e. Suppose f is an edge inG with weight strictly smaller than all other edges. Prove that every minimum spanning tree (MST)must contain f.

Let G' be the graph obtained by adding the edge (u, v) to G. Suppose that there exists a cycle C in…

Answered: (a) Suppose G = (V, E) is any…

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(a) Suppose G = (V, E) is any undirected graph and (S, V-S) is a cut with zero cut edges in G. Prove that if we pick two arbitrary vertices u ES and v EV - S, and add a new edge (u, v), in the resulting graph, there is no cycle that contains the edge (u, v).