We know that all trees are bipartite. Suppose the vertices of the tree when converting to a bipartite graph belong to the two sets A and B. It is not important that the number of vertices in the two sets are same. Also, it's not important that it is possible to reach every vertex in A from any vertex in B. Suppose we need to build a kind of Complete bipartite graph, in the sense that we can reach any vertex in A from B or reach any vertex in B from A and that the number of vertex in A is equal to B. Obviously, this is fairly simple by adding vague edges in the tree, which will fulfill our purpose but definitely it won't be a tree now because of the cycles created.
Let's suppose we are allowed to destroy the tree, but not by adding vague edges but can only add edges of the smallest length possible. That means, if we can get such a Complete Bipartite graph by adding an edge of length $3$ between one or maybe more vertex, then I may add edges of length less than $3$ or equal to $3$ but not more than that, because it's unnecessary. Unit length here would represent two adjacent vertex distance. So, if I cross two vertexes when I move from vertex $x$ to vertex $y$ then the length between $x$ and $y$ is $3$.
Now, about how I'm thinking: I think we just need to create one vertex in A that has an edge to every vertex in B and having one vertex in B that has an edge to every vertex in A. This way it's possible to reach any vertex in any set. It's obvious that we need to add some edges to keep equal number of elements in set A and B, but by the way I just stated, I think that will require adding least number of edges and hence maybe helpful in adding smaller edges altogether. Correct me, if I am wrong but I am unable to decide how to begin with this.
P.S. Complete Bipartite graph is just a name used here and has nothing to do with the term's original meaning.
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