For the given independence polynomial $1+9x+18x^2+9x^3$, find out all the single pendant vertex graphs with the given independence polynomial.
My approach is as follows:
Let $v$ be a single pendant vertex such that $v\sim v_1$. Now,
$2\leqslant \deg(v_1)\leqslant 6$. But by doing this manually, I have to consider a lot of cases. So, I am asking if there is any theorem which can be used or any method which can be applied to find all such graphs.