Defective chromatic polynomials

For a graph $G$ and an integer $d\geq 0$, the defective chromatic polynomial $χ_d(G;k)$ counts the $k$-colorings of $G$ in which each vertex has at most $d$ neighbors of its own color. We investigate which structural properties of $G$ are determined by the full family $\{χ_d(G;k)\}_{d\geq 0}$. We es