those chains. Right: The usual labelling on the product of chains showing $\cD(P)$ as $\cP - {[{{1}_{[2]}}, {{0}^{[1]}}]} - {[{{2}_{[1]}}, {{0}^{[3]}}]}$.[]{data-label="fig:PI"}](./DS2a.pdf "fig:"){width="8cm"}]{} (8.1,-0.1) (9.7,.4) (10.9,1.5) (11.9,2.6) (11.9,3.7) (8, .7) (10, 4.4) (7.2, 2.0) (7.2, 2.0) (0,0)[![Left: The lattice $\cD(P)$ from Figure \[fig:GPA\] embedded in a product of three chains, and the graph $G(P,A)$ from Figure \[fig:GPA\] embedded as an induced subgraph of the product of paths on those chains. Right: The usual labelling on the product of chains showing $\cD(P)$ as $\cP - {[{{1}_{[2]}}, {{0}^{[1]}}]} - {[{{2}_{[1]}}, {{0}^{[3]}}]}$.[]{data-label="fig:PI"}](./DS2b.pdf "fig:"){width="8cm"}]{} (2.7,-0.1) (4,.5) (4.9,1.5) (5.9,2.5) (5.9,3.7) (3,2.1) (2.8,3.1) (4.6,3.25) (4.8,2.5) (2.3, .9) (2.1, 2.1) (4, 4.4)

Our following, second, characterisation of reflexive $\operatorname{DL}$-graphs is immediate from Theorem \[thm:embedding\].

\[cor:charPI\] A reflexive graph $G$ is a $\operatorname{DL}$-graph if and only if it is the induced subgraph of a product $\cG = \prod_{i = 1}^dG_i$ of proper interval graphs $G_i$ in min-max form, that we get by removing vertex intervals.

A reflexive graph $G$ is $R$-thin if no two vertices have the same neighbourhood. For questions of $\operatorname{Ret}(G)$, one may always assume that $G$ is $R$-thin as there are simple linear time reductions between $\operatorname{Ret}(G)$ and $\operatorname{Ret}(G^R)$ where $G^R$, defined formally in Section \[sub:NRThin\], 