Q:

Ulam's Conjecture, Graph isomorphism, Application

Here is a short takeout on Ulam's Conjecture:

On the Wikipedia page it say it has been proven up to n=11.  
Now the Question: 
If I have to check for the Isomorphism, does this mean, if I take out an Vertex x in Graph one and then show that for all vertices in Graph two it is impossible to get G-x, then they could not be isomorphic?  As long I am have less the 11 edges.  Any suggestion on this? 

A:

If you take out a vertex from the first graph, leaving $G-x$, but you can't find any isomorphism between $G-x$ and $H-y$ for any $y \in V(H)$, then $G$ and $H$ are definitely not isomorphic. ( If there were an isomorphism $\phi$, then $H - \phi(x)$ would be isomorphic to $G-x$.)
Ulam's conjecture addresses the converse of this principle, and you don't need it to answer your question.