I have the following problem:
Let $H=(W, F)$ a graph and $k \in \mathbb{N^*}$ be an instance for problem $\textbf{CMP}$ (i.e. the clique problem). Let $W'$ a set of new vertices, $|W'|=|H|=n$. We define the following graph, $G=(V, E)$: $V=W \cup W'$, $E=\{xy | x, y \in V, xy \not\in F$ and $|W' \cap \{x, y\}| \leq 1\}$. Show that $H$ contains a clique of cardinality at least $k$ if and only if $G$ contains a subset of vertices $U$, $|U| \leq p=n-k$, such that $G-U$ is bipartite.
I understand that I have to prove the clique problem is polynomially reducible to the odd cycle transversal problem in order to solve this, but that's all I've got. My professor said this was actually very easy to prove, and I feel like that's true and I'm missing something obvious, but I can't quite figure out how these two problems are connected. I thought I could assume that $G$ contains odd-length cycles and go from there, but according to the definition above, $G$ is already a bipartite graph, which means I can't do that so now I'm completely out of ideas.