For an arbitrary graph $G$, one popular method for finding the number of spanning tress, $\tau(G)$, is Deletion-Contraction. This method uses the fact that $$ \tau(G) = \tau(G-e) + \tau(G\cdot e) $$ where $G-e$ denotes the graph obtained by deleting an arbitrary edge $e$, and $G\cdot e$ denotes the graph obtained by contracting an arbitrary edge $e$.
I am trying to use this method to find the number of spanning trees in the simple graph below.
I keep getting the wrong answer, and I suspect my problem lies in the ultimate case where I contract an edge, causing two vertices to be connected to each other by multiple edges.
When this happens, I have tried deleting the multiple edges, and I have tried keeping them. Either way, I get the wrong answer at the end.
The Wikipedia article on Spanning Trees states
In this formula, if the given graph G is a multigraph, or if a contraction causes two vertices to be connected to each other by multiple edges, then the redundant edges should not be removed, as that would lead to the wrong total.
Is this true? I have not been able to find any other sources that clarify this.