This is not an answer, but some context.

A somewhat related notion is that of <em>comparability graphs</em>: 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 <a href="https://en.wikipedia.org/wiki/Comparability_graph">Wikipedia article</a>.

You can find the list of forbidden graphs in <a href="https://web.archive.org/web/20230319195655/https://trotter.math.gatech.edu/math-3012/12-Cover_Graphs_and_Comparability_Graphs.pdf">these slides </a> (titled "Cover Graphs and Comparability Graphs") 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 <a href="https://cstheory.stackexchange.com/questions/7677/is-there-a-list-of-forbidden-subgraphs-for-comparability-graphs">this post</a> on CSTheory stackexchange.