it solves the problem because

The initial number of edges you start with is equal to the number of edges in the tree, which is equal to the number of vertices - 1, if the graph is connected.(1) Do a DFS traversal, generate the DFS tree.

Since the algorithm only decreases the number of edges, but never increases it, the number of edges will never be more than (or even equal) to the number of vertices.

So in your case above, the tree will have 3 edges ( (1,2) (2,3) (3,4) ), and the algorithm kicks out edge (2,3) and then terminates, so the final answer is ( (1,2) (3,4) ).