# Why are these two graphs isomorphic?

I have the following graphs and I am trying to identify if they are isomorphic or not.

The solutions that I have indicate they are isomorphic, with the following bijection:

However, it appears that the graph on the right has more 3-cycles than the graph on the left. The graph on the left also contains an obvious 4-cycle, where the graph on the right does not. The degrees of all the vertices are the same.

It's my understanding that two graphs cannot be isomorphic if they have a differing number of x-cycles. What am I missing here?

Think of this as pushing the vertices $f,b,d$ inside of the triangle $a,e,c$. There are plenty of four cycles in the second graph as well one is $a \rightarrow b \rightarrow d \rightarrow e \rightarrow a$