wheel ($V(G) = U \cup \{w\}$ and $E(G) = \{ (u_i, u_{i-1}), (u_i, w) ~:~ i \in [m] \}$), and *complete suns* $KS_m$, when $G$ is a complete graph of size $m$.

Let $m \geq 4$. Then, $\eta(CS_m) = \eta(WS_m) = 2$.

By Observation \[ADDITIVE1\], $\eta(CS_m) \geq 2$ and $\eta(WS_m) \geq 2$ so we only have to propose an additive 2-coloring of $CS_m$ and $WS_m$. We start with $CS_m$.

Consider an assignment $f:V \rightarrow \{1,2\}$ such that $f(u_i) = 2$ if $i$ is odd, $f(u_i) = 1$ if $i$ is even and $f(v) = 1$ for all $v \in V \backslash \{v_1\}$. If $m$ is even, also assign $f(v_1) = 1$. Thus, $f(N(u_i)) = 4$ if $i$ is odd, $f(N(u_i)) = 6$ if $i$ is even and $f(N(v)) = 3$ for all $v \in V$. If $m$ is odd, assign $f(v_1) = 2$. In this case, $f(N(u_1)) = 6$, $f(N(u_2)) = 7$, $f(N(u_m)) = 5$ and for $i = 3,\ldots,m-1$, $f(N(u_i)) = 4$ if $i$ is odd and $f(N(u_i)) = 6$ if $i$ is even. In addition, $f(N(v_m)) = 4$ and $f(N(v)) = 3$ for all $v \in V \backslash \{v_m\}$. Therefore, $f$ is an additive 2-coloring of $CS_m$.

For 