Check if edge is included in SOME MST in linear time (non-distinct values)

Question

I am working on an algorithm to check if a given edge is included in one of all possible mst's.

For this question, we are considering non-distinct values and our edge e connects vertices A & B.

So far, I have: If a path can be made from A to B consisting of edges with weights less than or equal to the weight of our edge e--we can say that edge e is not a part of any MST.

Am I missing anything here/ ideas on a better algorithm?

EDIT:

What are thoughts on a solution involving the cycle property-- So, we consider all edges with weight less than the edge we are considering. If we can make a path from A->B with those edges, we can say that it is not part of any MST?

Answer 1

We will solve this using MST cycle property, which says that, "For any cycle C in the graph, if the weight of an edge e of C is larger than the weights of all other edges of C, then this edge cannot belong to an MST."

Now, run the following O(E+V) algorithm to test if the edge E connecting vertices u and v will be a part of some MST or not.

Step 1

Run dfs from one of the end-points(either u or v) of the edge E considering only those edges that have weight less than that of E.

Step 2

Case 1 If at the end of this dfs, the vertices u and v get connected, then edge E cannot be a part of some MST. This is because in this case there definitely exists a cycle in the graph with the edge E having the maximum weight and it cannot be a part of the MST(from the cycle property).

Case 2 But if at the end of the dfs u and v stay disconnected, then edge E must be the part of some MST as in this case E is always not the maximum weight edge in all the cycles that it is a part of.