Find total number of ways to disconnect the following graph 
Find total number of ways to disconnect the following graph:




*

*$4$

*$5$

*$6$

*$8$



My attempt:
I've done manually to find possible disconnected sets  of given graph. I guess it is should be $8$.

Can you explain in formal way, please?

 A: The minimum number of edges that need to be removed to disconnect the graph is $3$, and there are just $4$ ways to do this: remove all of the edges incident at one vertex of degree $3$. There are $9$ ways to remove a set $E$ of edges so that (a) the resulting graph is disconnected, and (b) removing any proper subset of $E$ leaves a connected graph. However, $5$ of these require the removal of $4$ edges, not $3$:
$$\begin{align*}
&ac,bc,dc,ed\\
&ad,ac,cb,be\\
&ad,dc,ce,eb\\
&ab,bc,ce,ed\\
&ba,ac,cd,de
\end{align*}$$
To verify that there really are only $4$ possibilities when you remove just $3$ edges, suppose that you remove $3$ edges, but not all $3$ of the edges incident at any of the corner vertices. 


*

*If all three of the removed edges are incident at $c$, it’s easy to see that the resulting graph is still connected.  

*If two of them are incident at $c$, removing those $2$ leaves a graph consisting of two cycles that share either one or two edges, depending on which two edges incident at $c$ were removed, and removing one more edge cannot disconnect such a graph.  

*If one of them is incident at $c$, without loss of generality suppose that it’s $ac$. Then at least one of $ab$ and $ad$ was not removed; without loss of generality we may assume that $ab$ is still present. The edges $ab,cd,ce$, and $cb$ are then all present and are sufficient to connect the graph.  

*If none of them is incident at $c$, the $4$ edges incident at $c$ are sufficient to connect what remains.

