For any finite poset $P=(X,\leq)$ there is an undirected graph $G$ underlying the Hasse diagram of $P$ such that $V(G)=X$ and $E(G)=\{\{u,v\}:u\lessdot v\}$. With that said, is it possible to characterize all these graphs? That is, those graphs which underlie the Hasse diagram of some finite poset?

Clearly they must all be triangle free since any orientation of a triangle will result in either a cycle which isn't acyclic or an arc between two vertices and a path of length two between those vertices which isn't transitively reduced. Moreover its easy to see any bipartite graph will fall into this class since if the vertices in an arbitrary graph $G$ can be partitioned into two independent sets $X$ and $Y$ then if we define $R=\bigcup_{e\in E(G)}(e\cap X)\times (e\cap Y)$ we see $P=(X\cup Y,R)$ is a poset with a height of one and that $G$ underlies the Hasse diagram of $P$. In fact if for any poset $P$ we let $h(P)$ denote the height of $P$ then whenever a graph $G$ underlies the Hasse diagram of $P$ we must have that $\chi(G)\leq h(P)+1$ as a consequence of the Gallai–Hasse–Roy–Vitaver theorem. Though despite this class of graphs being closed under edge subdivisions, graphs in this class are not closed under edge contractions which rules out using the graph minor theorem. However because this class of graphs is closed under the removal of edges and vertices (since removing arcs from the Hasse diagram of any finite poset yields another Hasse diagram which is a subposet of its parent) it would still make sense to look for some other type of forbidden subgraph characterization (not necessarily involving minors) of this class. In fact one can prove if we call any digraph a psuedocycle when it is either a directed cycle or if it can be formed by flipping one arc in a directed cycle, then there exists a family of pairwise non-isomorphic graphs $\mathcal{F}$ where $G\in\mathcal{F}$ iff the following holds:

$$(1)\text{ Every orientation of }G\text{ contains at least one psuedocycle}$$ $$(2)\text{ Every proper subgraph of $G$ has an orientation containing no psuedocycles}$$

Such that any graph can be oriented to form a Hasse diagram of a finite poset iff it has no subgraph isomorphic to an element in $\mathcal{F}$. But how can this class of forbiden subgraphs be further simplified? Now I can prove if $G$ is one of these forbiden subgraphs then $G$ must be $2$-vertex connected and satisfy $\chi(G)\geq \text{girth}(G)$ though this alone still hardly narrows down what exactly these forbidden subgraphs look like. Is it possible to simplify the above criteria for these forbiden subgraphs? I know the smallest two by order are the triangle graph and the Grötzsch graph, but what are some others?