following result allows to solve the ACP of a join of $G$ with a complete graph by just solving the ACP of $G$:

\[UNIVERTICES\] Let $G$ be a graph of $n$ vertices and $\Delta$ be the largest degree in $G$. Then, $\eta(G \lor K_q) = \max \{ \eta(G), q \}$ for all $q \leq n-\Delta-1$.

Let $V$ and $E$ be the set of vertices and edges of $G$ respectively, $U = \{ u_1, u_2, \ldots, u_q \}$ be the set of vertices of $K_q$, $G' = G \lor K_q$ and $f$ be an optimal additive coloring of $G$. Consider a labeling $f'$ of $G'$ satisfying $f'(v) = f(v)$ for all $v \in V$, and $f'(u_i) = i$ for all $i \in [q]$. Now, for any $(v,v') \in E$, $f'(N_{G'}(v)) = f(N_G(v)) + f'(U) \neq f(N_G(v')) + f'(U) = f'(N_{G'}(v'))$. For any $i,j \in [q]$ such that $i < j$, $f'(N_{G'}(u_i)) = f(U \cup V) - i > f(U \cup V) - j = f'(N_{G'}(u_j))$. Finally, note that $f'(V \backslash N_G(v)) \geq n - d_G(v) \geq n - \Delta$ for all $v \in V$. Then, for any $u \in U$ and $v \in V$, $f'(N_{G'}(u)) = f'(U \cup V) - f'(u) 