of external vertices $O_d= \{a^v_1,b^v_1,a^v_2,b^v_2,\ldots,a^v_d,b^v_d\}$ and internal vertices $M_d$. The internal vertices form a copy of the set of those 0-1-strings of length $d$ that have an even number of entries equal to 1. For each $i$, each internal vertex $m$ is adjacent to exactly one vertex of $\{a^v_i,b^v_i\}$, namely it is adjacent to $a_i$ if the $i$-th bit of the string $m$ is 0 and to $b_i$ otherwise. An example of $F_3$ is depicted in Figure \[fig:cfi\]. It remains to explain how the different gadgets are interconnected. For this, for a vertex $v\in\Gamma$ of degree $d$ each edge is associated with one of the pairs $a^v_i,b^v_i$. For an edge $(u,v)\in E(\Gamma)$ assume $u$ is associated with the pair $(a^u_i,b^u_i)$ in the gadget corresponding to $u$ and $v$ is associated with the pair $(a^v_j,b^v_j)$ in the gadget corresponding to $v$. Then we insert (parallel) edges $\{a^u_i,a^v_j\}$ and $\{b^u_i,b^v_j\}$. Adding such parallel edges for each edge of the base graph we obtain the graph $\operatorname{CFI}(\Gamma)$. The *twisted CFI-graph* $\widetilde{\operatorname{CFI}(\Gamma)}$ is obtained by replacing one pair of (parallel) edges $\{a^u_i,a^v_j\}$ and $\{b^u_i,b^v_j\}$ with the (twisted) edges $\{a^u_i,b^v_j\}$ and $\{b^u_i,a^v_j\}$. It can be shown that for connected base graphs (up to isomorphism) it 