I have the following question:
Suppose we have a finite graph $G=(V,E)$. Now take two arbitrary spanning sub-graphs, i.e. $G_1 = (V,E_1)$ and $G_2=(V,E_2)$ with $E_1,E_2 \subseteq E$. Suppose we would like to transform $G_1$ into $G_2$ by first deleting all edges in $E_1-E_2$ from $E_1$ and then inserting all edges from $E_2-E_1$.
So our transformation $G_1 \rightarrow G_2$ consists of a chain of simple transitions, which are either edge-deletions or edge-insertions.
Now fix such an arbitrary transition. What is the number of spanning subgraphs $G_1$ and $G_2$ for which we would need to use this transition for the transformation $G_1 \rightarrow G_2$?