_0^*}=\{P_0,$[$S_{0,0}$]{},[$S_{0,0,0}$]{}$\}$ (these stars have Grundy value 0);  -   ${\mathcal C_0^*}=\{P_0$,[$S_{0,0}$]{}$\}$ (these stars have Grundy value 0);  -   ${\mathcal C_0^{\Box}}$: subdivided stars $S$ such that ${\mathcal{G}}(S)=0$ and $S$ contains one or three paths of length $0$;  -   ${\mathcal C_0^{\Box}}$: subdivided stars $S$ such that ${\mathcal{G}}(S)=0$ and 