Let $G=(V,E)$ be a simple, undirected graph. Is there a partial ordering $\leq\subseteq (V\times V)$ with the following property? $$\{v,w\} \in E \text{ if and only if } v||y$$

(We write $v||w$ in the poset $(V,\leq)$ if $v\not \leq w$ and $w\not\leq v$?)