this paper we give two explicit characterisations of the class of reflexive $\operatorname{DL}$-graphs, and use these characterisations to address the problem of recognition.

For our first characterisation, we recall a well known result of Birkhoff [@Bi]. For a poset $P$, a subset $D$ is a [*downset*]{} if $b \in D$ and $a \leq b$ implies $a \in D$. The family $\cD(P)$ of all downsets of $P$ is a distributive lattice under the ordering $\subseteq$. The meet and join operations are $\cap$ and $\cup$, respectively. Birkhoff showed that for any distributive lattice $L$, $L$ is isomorphic to $\cD(J_L)$ for a unique poset $J_L$, (the poset of join irreducible elements of $L$).

Viewing a comparability $a \leq b$ as an arc $(a,b)$, a poset $P$ is just a transitive acyclic (except for loops) reflexive digraph. So we can talk of a sub-digraph $A$ of $P$.

\[def:GPA\] For a poset $P$ and a sub-digraph $A$ of $P$, let $G = G(P,A)$ be the graph on $\cD(P)$, in which two downsets $D, D' \in \cD(P)$ are adjacent if $A$ contains all arcs $(x,y)$ of $P$ for which $x$ and $y$ are in either $D {-}D'$ or $D' {-}D$.

See Figure \[fig:GPA\] for an example. 