I have a question regarding the general method of determining whether two graphs are isomorphic to one another.
I understand that for small graphs one may look at a graph and determine rather easily by inspection whether or not two graphs are isomorphic.
I also understand that a way to show that two graphs are not isomorphic is to show that one graph $G_1$ has an invariant property while another graph $G_2$ does not possess the same property. From what I understand, typically one shows that a graph is not isomorphic to another by utilizing invariant properties.
My question is whether showing two graphs possess invariant properties is enough to show that they are isomorphic. For example, if you show that two graphs possess two simple 3-cycles, does this automatically guarantee they are isomorphic?