Consider points randomly distributed in space. The goal is to find the graph that has the smallest Euclidean length that connects these points. The answer is clearly the minimum spanning tree which has edges. Now suppose we are required to use exactly edges and find the smallest length graph . Clearly, will contain exactly one cycle. Now suppose we remove the longest edge from this cycle and obtain graph . I am wondering if for any configuration of points. My intuition is that it is true, and I'm not able to find a counter example. However, I am not able to prove it. Can someone help?