One of the basic motivations behind the Minimum Spanning Tree Problem is the goal of designing a spanning network for a set of nodes with minimum total cost. Here we explore another type of objective: designing a spanning network for which the most expensive edge is as cheap as possible.

Specifically, let G = (V , E) be a connected graph with n vertices, m edges, and positive edge costs that you may assume are all distinct. Let T = (V , E′) be a spanning tree of G; we define the bottleneck edge of T to be the edge of T with the greatest cost.

A spanning tree T of G is a minimum-bottleneck spanning tree if there is no spanning tree T′ of G with a cheaper bottleneck edge.

(a) Is every minimum-bottleneck tree of G a minimum spanning tree of G? Prove or give a counterexample.

(b) Is every minimum spanning tree of G a minimum-bottleneck tree of G? Prove or give a counterexample.