we can get a vertex $v_2 = \phi\hspace{0.5mm} u_2$ of $H$ so that there is a bijection between $\Pi_{u_1,u_2}^*$ and $\Pi_{v_1,v_2}^*$ induced also by $\phi$, where $\Pi_{u_1,u_2}^*$ and $\Pi_{v_1,v_2}^*$ are two partitions of $V(G)$ and $V(H)$, respectively, consisting of orbits of $( {\mathrm{Aut} \hspace{0.4mm} G} )_{u_1,u_2}$ and of $( {\mathrm{Aut} \hspace{0.4mm} H} )_{v_1,v_2}$, and $( {\mathrm{Aut} \hspace{0.4mm} G} )_{u_1,u_2}$ called the [*stabilizer of the sequence $u_1,u_2$ in ${\mathrm{Aut} \hspace{0.4mm} G}$*]{} is defined as $\{ \gamma \in {\mathrm{Aut} \hspace{0.4mm} G} : \gamma \hspace{0.5mm} u_i = u_i, i = 1,2 \}$. Clearly, we can continue this process until the stabilizer of the sequence $u_1,\ldots,u_s$ is trivial, [*i.e.,*]{} $( {\mathrm{Aut} \hspace{0.4mm} G} )_{u_1,\ldots,u_s} = \{ I \}$.

Conversely, if we have those two groups of partitions $\Pi_{G}^*,\Pi_{u_1}^*,\ldots,\Pi_{u_1,\ldots,u_s}^*$ and $\Pi_{H}^*,\Pi_{v_1}^*,\ldots,\Pi_{v_1,\ldots,v_s}^*$ and know the corresponding relations between cells of partitions in each pair $(\Pi^*_G,\Pi^*_H)$, $(\Pi^*_{u_1},\Pi^*_{v_1})$, $\cdots$, $(\Pi^*_{u_1,\ldots,u_s},\Pi^*_{v_1,\ldots,v_s})$, then we can easily decide whether $G$ is isomorphic to $H$ or not and in the case of being isomorphic work out an isomorphism from $G$ to $H$.

In the next part, we shall explore some geometric features of $\Pi_G^*$ that show us how to reduce the problem of determining $\Pi_G^*$ to that of working out a series 