How to find Graph that is not isomorphic?

Question

I need to find Graph G with degree sequence (5,5,4,4,4,4,4,2,2,2,2,2) constructed with Havel Hakimi method, that will not be isomorphic with G. Is it even possible?

Answer 1

Using the Havel-Hakimi Theorem we can build graph $H$. After some thought I came up with graph $G$, such that: $G \ncong H$. Why?

In graph $G$, we have: $v_{i}$ adjacent to two vertices of degree $2$ and a vertex of degree $5$, for $i = 1, 2$. Where as in graph $H$, we can clearly see that vertex $u_{2}$ doesn't have such a configuration.

Furthermore, note that both graphs have degree sequence; $S: 5, 5, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2$.