I am given a graph $ G = (V, E)$ with $ N$ connected components and $ G^\prime = (V^\prime, E^\prime)$ , where for each $ v \in V$ there is $ v_1, v_2 \in V^\prime$ and for each $ (u, v) \in E$ there is $ (u_1, v_2), (u_2, v_1) \in E^\prime$ .

I need to prove:

$ $ G^\prime \textrm{ has 2N connected components} \Leftrightarrow G \textrm{ is bipartite}$ $

I don’t know what technique or approach I need to use to prove this in either direction. I have a feeling I can use a direct proof for both directions. I have tried to come up with something but after an hour nothing has come up.