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abstract: |
    Let $G$ and $H$ be two simple graphs. A bijection $\phi:V(G)\rightarrow V(H)$ is called an [*isomorphism*]{} between $G$ and $H$ if $(\phi\hspace{0.5mm} v_i)(\phi\hspace{0.5mm} v_j)\in E(H)$ $\Leftrightarrow$ $v_i v_j\in E(G)$ for any two vertices $v_i$ and $v_j$ of $G$. In the case that $G = H$, we say $\phi$ an automorphism of $G$ and denote the group consisting of all automorphisms of $G$ by $\mathrm{Aut}\hspace{0.5mm} G$. As well-known, the problem of determining whether or not two given graphs are isomorphic is called [*Graph Isomorphism Problem*]{} (GI). One of key steps in resolving GI is to work out the partition $\Pi^*_G$ of $V(G)$ composed of orbits of ${\mathrm{Aut} \hspace{0.4mm} G}$. By means of geometric features of $\Pi^*_G$ and combinatorial constructions such as the multipartite graph ${\left[\Pi^*_{t_1},\cdots,\Pi^*_{t_s}\right]}$, where $t_1,\ldots,t_s$ are vertices of $G$ constituting an orbit of ${\mathrm{Aut} \hspace{0.4mm} G}$ and $\Pi^*_{t_i}$ ($i=1,\ldots,s$) is the partition comprised of orbits of the stabilizer $( \mathrm{Aut}\hspace{0.5mm} G )_{t_i}$, we can reduce the problem of determining $\Pi_G^*$ to that of working out a series of partitions of $V(G)$ each of which consists of orbits of a stabilizer that fixes a sequence of vertices of $G$, and thus the determination of the partition $\Pi^*_v$ 