$\Gamma_G$. Our working example will be the following construction but other choices are certainly possible and many choices will lead to equivalent or at least related algorithms.

To obtain the graph $\Gamma_G$ (see Figure \[fig:mult:gadet\]), start with a set of isolated nodes corresponding to elements of the group $G$. For each pair of group elements $(g,h)$ add a multiplication gadget $M(g,h)$ by adding 4 nodes $a_{gh},b_{gh},c_{gh},d_{gh}$ and add the edges $E(M(g,h))=$ $$\{\{g,\!a_{gh}\},\{h_{gh},\!b_{gh}\},\{gh,\!d_{gh}\},\{a_{gh},\!b_{gh}\},\{b_{gh},\!c_{gh}\},\{c_{gh},\!d_{gh}\}\}.$$

We then use the classical $k$-dimensional WL-algorithm on the graph $\Gamma_G$ and pull back the colorings of $k$-tuples by simply restricting it to $G^k$.

By construction we have $|\Gamma_G|=\Theta(|G|^2)$ and due to vertex-degrees $G$ is a canonical subset of the vertices of $\Gamma_G$. Thus for two groups $G,H$ we have $\Gamma_G\cong \Gamma_H$ if and only if $G\cong H$.

Many other reductions of this form transforming groups to graphs are possible. However, some of them are artificial. For example, one could artificially ensure that the resulting graph has low WL-dimension by precomputing certain isomorphism-invariants not captured by the WL-algorithm. Thus, a unified treatment of all isomorphism-preserving constructions seems infeasible. However, it seems that for many ’well-behaved’ functors the WL-dimension of the constructed graphs differ by a constant factor only. 