This is a homework question. I do not want the solution - I'm offering the solution I've been thinking of and wish to know whether is it good or why is it flawed.
My motivation is to find what edges of an unweighted, undirected graph are not a part of any MST. This problem only makes sense when several edges have the same values, otherwise the MST is unique.
My idea comes from Prim's Algorithm with a slight change - instead of adding the minimum edge from S to T on every step (where S and T being the two sets of vertex) - instead look for the minimum edge and more edges of the same value going from S to the vertex the minimum edge goes to. By doing that, (so I suppose) we will receive a graph containing all the edges which appear in any MST. If this is right, I can simply XOR the edges list with the original graph edges list to find what edges are not in any MST.
Thanks in advance.
Do you add all the edges you find (=those with equal weight)? If so, you will lose some edges:
Consider a pentagon with equal edge costs. You start with 1 node and add the 2 edges to the 2 adjacent nodes. In you next step you would add the 2 edges going from those 2 adjacent nodes to the 2 disconnected nodes and you would be done. However, all edges are of equal cost and they are all valid to be in the MST. The edge between the last 2 nodes is not included by your algorithm but could be part of the MST.
It's even worse. Suppose that last edge is of lower cost. Your algorithm still doesn't include it, yet it's present in every MST. You're adding several edges per step to account for all the possibilities but adding those edges changes the next steps.