is the $R$-thin graph we get from $G$ by removing all but one vertex from every set of vertices sharing the same neighbourhood. In Section \[sect:recog\] we prove the following.

\[thm:poly\] There is a polynomial time algorithm to decide whether or not an $R$-thin reflexive graph is a $\operatorname{DL}$-graph.

In fact, we give a polynomial time algorithm that not only decides if a given $R$-thin graph has a compatible distributive lattice, but if it does, finds one (actually all) such lattices.

While for questions about $\operatorname{Ret}(G)$ one may assume that $G$ is $R$-thin, distributive lattice polymorphisms are unusual in the fact that the existence of a compatible distributive lattice for $G^R$ does not imply the existence of one for $G$. We finish of Section \[sect:recog\] with some notes about deciding if a non $R$-thin graph is a $\operatorname{DL}$-graph.

Definitions, Notation, and Basic Observations {#sec:def}
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For any element $u$ of any ordering, ${\bm{\langle} u ]}$ is the set of elements below $u$, and ${[ u \bm{\rangle}}$ is the set of elements above it. We write $a \prec b$ if $b$ [*covers*]{} $a$; that is, if $a < b$ and there is no $x$ such that $a < x < b$. 