Consider the following divide$-$and$-$conquer approach to minimum spanning trees: Given a graph G $=($V$,$E$),$ divide the set V into V$1$ and V$2$ such that the size of V$1$ and V$2$ differ by at most one. Let E$1$ be the set of edges only incident on V$1,$ and E$2$ be the set of edges only incident on V$2.$ Then, recursively solve the minimum spanning tree problem on $($V$1,$E$1)$ and $($V$2,$E$2).$ Finally, select the lightest edge in E crossing the cut $($V$1,$V$2).$ The result is the minimum spanning tree. Prove or disprove that this correctly finds the minimum spanning tree of G$.$