The reduction function given by Richard Karp in 'Reducibility among combinatorial problems' for Directed Hamiltonian Cycle $\leq_{p}$ Undirected Hamiltonian Cycle goes as follows :
for input $G = (V,E)$, return $G' = (N, A)$ where
$N = V $ x $ \{0,1,2\}$ $A = \{ \hspace{5pt} \{ \langle u,0 \rangle,\langle u,1\rangle \}, \{ \langle u,1\rangle, \langle u,2\rangle \}\hspace{5pt} | \hspace{5pt} u \in V \} \cup \{ \hspace{5pt} \{ \langle u,2 \rangle,\langle v,0\rangle \} \hspace{5pt} | \hspace{5pt} \langle u,v \rangle \in E \} $
Why do we need the $\langle u, 1 \rangle$ vertices between $\langle u, 0 \rangle$ and $\langle u, 2 \rangle$ ?
Thanks!