Professor Borden proposes a new divide-and-conquer algorithm for computing minimum spanning trees, which goes as follows. Given a graph G = (V, E), partition the set V of vertices into two sets V₁ and V₂ such that |V₁| and |V₂| differ by at most 1. Let E₁ be the set of edges that are incident only on vertices in V₁, and let E₂ be the set of edges that are incident only on vertices in V₂. Recursively solve a minimum-spanning-tree problem on each of the two subgraphs G₁ = (V_1, E₁₎ and G₂ = (V_2, E₂₎. Finally, select the minimum-weight edge in E that crosses the cut (V_1, V₂), and use this edge to unite the resulting two minimum spanning trees into a single spanning tree.

Either argue that the algorithm correctly computes a minimum spanning tree of G, or provide an example for which the algorithm fails.