Let G = (V, E) be a loop-free connected undirected graph. Let H be a subgraph of G. The complement of H in G is the subgraph of G made up of those edges in G that are not in H (along with the vertices incident to these edges).
(a) If T is a spanning tree of G, prove that the complement of T in G does not contain a cut-set of G.
(b) If C is a cut-set of G, prove that the complement of C in G does not contain a spanning tree of G.