# Is every graph an incomparability graph?

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 $$w\not\leq v$$?)

Any incomparability graph 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.