title: Given a graph G, does a divide-and-conquer approach work to finding minimum spanning trees?
body: <p>If given a connected graph G, split the graph into Ga and Gb. If you find a minimum spanning tree for Ga and Gb (called Xa and Xb, respectively), does connecting Xa to Xb with a minimal weighted edge still form a spanning tree? Is that spanning tree a minimum spanning tree?</p>
  
  <p>This is my logic so far. I believe that connecting Xa to Xb <em>would</em> form at least a spanning tree almost by definition. (If there is a counterxample though that'd be helpful) However, I don't think it would always form a minimum spanning tree because depending on the structure of the graph, you may be able to remove an edge from Xa or Xb, then add the edge connecting them, and still have a tree. This may be the case in a situation where multiple edges of the same weight connect Xa and Xb at different vertexes. </p>
  
  <p>Is my logic correct so far?</p>