is irrelevant which edge is twisted [@DBLP:journals/combinatorica/CaiFI92]. For a subset of the edges of the base graph $E'\subseteq E(\Gamma)$, we can define the graph obtained by twisting exactly the edges in $E'$. The resulting graph is isomorphic to $\operatorname{CFI}(\Gamma)$ if $|E'|$ is even and isomorphic to $\widetilde{\operatorname{CFI}(\Gamma)}$ otherwise.

In the original construction the base graph is usually thought of as vertex colored with all vertices obtaining a different color. This makes all gadgets distinguishable. The colors can be removed by attaching gadgets retaining the property that the base graph is identified by 2-dimensional Weisfeiler-Leman. We want to record here the observation that it is possible to choose the base graph of WL-dimension 2 while maintaining the property that it is 3-regular.

\[obs:base:graph:low:WL:dim\] The 3-regular base graph $\Gamma$ can be chosen to have Weisfeiler-Leman dimension at most 2.

This can be seen in two ways, by adding gadgets on edges or by observing that random expanders, usually used in the construction, have this property.

First order logic with counting {#subsec:first:order:logic:graphs}
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There is a close connection between the Weisfeiler-Leman algorithm of dimension $k$ and the $(k+1)$-variable fragment of first order logic on graphs with counting quantifiers [@DBLP:journals/combinatorica/CaiFI92]. To obtain this logic 