Recall that the Minimum-Spanning-Tree problem takes as input an undirected graph G=(V,E) with edge weights w:ER+and outputs a subset of edges TE such that (V,T) is a spanning tree and the weight eTw(e) is minimized. Now consider the related Minimum-Bottleneck-Tree problem where instead of minimizing the spanning tree's total weight, we output a spanning tree TminimizingmaxeTw(e), the weight of its most expensive edge. (a) Is every minimum bottleneck tree of G also a minimum spanning tree of G ? If yes, prove it; if no, provide a counterexample. (b) Is every minimum spanning tree of G also a minimum bottleneck tree of G ? If yes, prove it; if no, provide a counterexample. [Hint: If you're claiming the answer is "yes", you need to start your proof with something like "Consider any undirected graph G=(V,E) and any minimum spanning/bottleneck tree TE. " The rest of the proof is then you arguing why T is also a bottleneck/spanning tree for G. If you're claiming the answer is "no", you need to define a specific graph G=(V,E)-i.e., tell me what its nodes and edges are-identify a minimum spanning/bottleneck tree TE in G, and show that T is not also a minimum bottleneck/spanning tree.]