Let $G=(V,E)$ be a simple, undirected graph. Is there a [partial ordering](https://en.wikipedia.org/wiki/Partially_ordered_set) $\leq\subseteq (V\times V)$ 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 $w\not\leq v$?)



Is every graph an incomparability graph?
Any [incomparability graph](https://www.sciencedirect.com/science/article/pii/S0001870812001077) is perfect (shown by Dilworth in 1950), so any non-perfect graph will be a counterexample. For an explicit counterexample, choose the cycle on $5$ vertices.


