Removing edges to eliminate all but one cycle-free path between two nodes

I have a connected undirected graph having n nodes. Given two nodes, I want to find the minimum number of edges that would have to be removed in order to ensure that there's only one cycle-free path between those two nodes.

For example, if this is the graph:

1------------2------------5
|                         | 
|                         | 
3-------------------------4

then given the nodes 1 and 5, the answer will be 1: just remove (for example) the edge between node 3 and node 4.

The brute-force approach is, for each subset of the set of edges, to try removing those edges and test if there's a unique cycle-free path between the two nodes of interest.

Is there a more efficient approach? (I Googled it, but did not find anything relevant.)

(Dear cryptomanic, I added these examples to help in the discussion about the exact requirements; please edit this part and indicate which of these solutions are valid. m69)

Input graph: (going from X to Y)

          O---O---O---O       O
         /     \ /     \     / \
O---O---X       O       Y---O---O
         \             /
          O---O---O---O
             / \       \
            O---O       O

Solution A: (no cycles inbetween X and Y)

          O---O---O---O       O
         /       /     \     / \
O---O---X       O       Y---O---O
                       /
          O---O---O---O
             / \       \
            O---O       O

Solution B: (no side-paths inbetween X and Y)

          O---O---O---O       O
         /             \     / \
O---O---X       O       Y---O---O
                       /
          O---O---O---O
             / \       \
            O---O       O

Solution C: (no cycles connected to X and Y)

          O---O---O---O       O
         /       /     \       \
O---O---X       O       Y---O---O
                       /
          O---O---O   O
             / \       \
            O---O       O

Solution D: (completely isolate path from X to Y)

          O---O---O---O       O
         /             \     / \
O---O   X       O       Y   O---O

          O---O---O---O
             / \       \
            O---O       O

Solution E: (P can only be used once, so P-Q-R-P is not part of an alternative path)

          O---O---O---O       O
               \ /           / \
O---O---X       O       Y   O---O
         \             /
          O---P---O---O
             / \       \
            Q---R       O

Solution F:

          O---O---O---O       O
               \ /     \     / \
O---O---X       O       Y---O---O
         \             /
          O---O---O---O
             / \       \
            O---O       O