Find all solution to the deck of following card
there is 7 cards, so I know that the graph $G$ has order $n=7$. Let $m_i$ be the size of $G-v_i$ for $1 \leq i \leq 7$ then
$$m=\frac{\sum_{i=1}^n m_i}{n-2}=\frac {30} 5=6$$
$deg(v_i)=m-m_i$ for $1 \leq i \leq 7$ then we have the degree of vertices in order $1,1,1,2,3,1,3$. the graph I got is like $G_1$ with a vertex between 2 vertices of degree 3. Is there any other solution for this ?
If you add a vertex by an edge to a vertex of degree 1 in $G_1$, you have a solution to your problem. The one you describe, with a vertex of degree 2 between two vertices of degree 3 can not be obtained, for example, by appending a vertex to $G_6$.