The left side shows a poset $P$ represented by its Hasse diagram (defined in Section \[sec:def\]) in thick light edges, and a sub-digraph $A$ in dark edges, missing only the arc $(b,c)$. On the right is the downset lattice $\cD(P)$ again represented by its Hasse diagram, and the graph $G(P,A)$ missing only edges between vertices one of which contains $b$ and $c$ and the other of which contains neither of them.

(8,5)(0,0) (0,0)[![Poset $P$ and lattice $\cD(P)$ in thick light edges. Digraph $A$ and (the complement of) graph $G(P,A)$ in dark. []{data-label="fig:GPA"}](./DS1.pdf "fig:"){width="8cm"}]{} (1.3,0) (1.1,1.3) (2.6,1.3) (1.3,3.2) (2.7,3.2) (4.2,0) (5.6,0.4) (4.2,1.1) (6.3,1.1) (5.8,2.4) (7.6,2.4) (4.0,3.7) (6.3,3.7) (5.2,4.7)

Our first main theorem, Theorem \[thm:char2\] says that for any $\operatorname{DL}$-pair $(G,L)$, $G$ is isomorphic to $G(J_L,A)$ for some sub-digraph $A$ of $J_L$. Showing, in Lemma \[lem:pographiscompat\] that $G(P,A)$ is always a $\operatorname{DL}$-graph, we get the following characterisation of reflexive $\operatorname{DL}$-graphs.

\[cor:char2\] A reflexive graph $G$ is a $\operatorname{DL}$-graph if and only if there is there is a poset $P$ and a sub-digraph $A$ such that $G {\buildrel {} \over \cong}G(P, A)$.

In [@Ga] it was shown that a graph (without loops) is a [*proper interval graph*]{} if and only if there is 