every vertex of $I$ with colour $2$, as doing this will not create a $2$-cycle. If $|J|\geq 2$ then, in order to avoid a $2$-cycle, at least $|I|-1$ vertices of $I$ must be coloured $1$. This proves Claim \[clm:1\].

By \[prop:i\], \[prop:iii\], \[prop:iv\], we find that $S_2$ and $T_2$ are twin-sets. Let $Z$ be the set of isolated vertices in the subgraph of $G$ induced by $S_1\cup T_1$. Then by \[prop:i\], \[prop:ii\], \[prop:iv\], the neighbourhood of every vertex in $Z\cap S_1$ (respectively, $Z\cap T_1$) is $T_2\cup \{u\}$ (respectively, $S_2\cup \{u\}$). So $Z\cap S_1$ and $Z\cap T_1$ are twin-sets.

We choose one vertex from each non-empty set in $\{S_2,T_2,Z\cap S_1,Z\cap T_1\}$ and let $W$ be the set of chosen vertices. Note that the choice of the vertices in $W$ can be done arbitrarily, since all four of these sets are twin-sets. We now branch by giving all vertices in $S_2 \setminus W$ the same colour, all vertices in $T_2 \setminus W$ the same colour, all vertices in $(Z\cap S_1)\setminus W$ the same colour and all vertices in $(Z\cap T_1)\setminus W$ the same colour. We then branch by colouring the at most four vertices of $W$ with every possible combination of colours. 