$(16\%)$ This problem set is about minimum spanning trees. Sollin's algorithm is a minimum spanning tree. It works as follows.

Step $1$: Start with a forest that has n spanning trees $($each has one vertex$).$

Step $2$: Select one minimum cost edge for each tree. This edge has exactly one vertex in the tree.

Step $3$: Delete multiple copies of selected edges and if two edges with the same cost connecting two trees, keep only one of them.

Step $4$: Repeat until we obtain only one tree.

$($a$)(4\%)$ Given a connected weighted graph G$,$ is minimum spanning tree of G unique? If YES, please justify you answer; If NO$,$ please provide a counterexample.

$($b$)(4\%)$ What is the worst$-$case time complexity of the Sollin's algorithm mentioned above? $($Please analyze step by step$).$

$($c$)(8\%)$ Please prove that Sollin's algorithm is able to find the minimum spanning tree.