I Circuits or edge-disjoint unions of circuits in a connected graph

[cianfa72]

TL;DR Summary

Evaluate the number of nonempty circuits or edge-disjoint unions of circuits in a connected graph

Hi,
I've a question related to the graph theory.

Take a connected graph with $n$ nodes and $b$ edges. We know there are $m = b - n + 1$ fundamental circuits.

Which is the total number of nonempty circuits or edge-disjoint unions of circuits ? If we do not take in account the circuit orientation I believe the answer is $2^m - 1$.

Is the above correct ? Thanks.

 

[cianfa72]

To be more specific consider the following graph with 2 nodes and 4 edges. We have 3 fundamental circuits, but which is the total number of circuits or edge-disjoint unions of circuits ?