This is not an answer, but some context.
A somewhat related notion is that of comparability graphs: these are the graphs for which there is a poset $P$ on the vertex set such that $\{x,y\}$ is an edge $\Leftrightarrow$ $x$ and $y$ are comparable in $P$. A forbidden subgraph characterization was given by Gallai in 1967. See the Wikipedia article.
You can find the list of forbidden graphs in these slides on Tom Trotter's page. I think at one point he also had some course notes posted on it; it's also in his book on dimension theory of posets. See also this post on CSTheory stackexchange.
