_0^*}=\{P_0$,[$S_{0,0}$]{}$\}$ (these stars have Grundy value 0);  -   ${\mathcal D_0^{\Box}}$: subdivided stars $S$ such that ${\mathcal{G}}(S)=0$ and $S$ contains one or three paths of length $0$;  -   ${\mathcal D_0^{\Box}}$: subdivided stars $S$ such that ${\mathcal{G}}(S)=0$ and $S$ contains one or three paths of length $0$;  -   For $i\in \{0,0,0,0\}$, $\mathcal 