with $x',y'$ in $D {-}D'$, we clearly have that $x,y$ are in $D {-}D'$ as well, and so $D \not\sim D'$ in $G(P,A)$ whether $x' \to y'$ or not.

For a sub-digraph $A$ of a poset $P$, let ${{A^{\rm c}}}$ be the sub-digraph of $P$ with arc set $$\{ (x, y)  \mid (x,y) \in P {-}A \}.$$ The following is a useful alternate definition of adjacency in $G(P,A)$.

\[lem:altGPA\] Where $A$ is a sub-digraph of a poset $P$, and $D$ and $D'$ are in $\cD(P)$, $D$ and $D'$ are adjacent in $G(P,A)$ if and only if the following hold for all vertices $x$ and $y$.

-   If $x \in D$ and $(y,x) \in {{A^{\rm c}}}$ then $y \in D'$, and

-   If $x \in D'$ and $(y,x) \in {{A^{\rm c}}}$ then $y \in D$.

The definition of adjacency of $D$ and $D'$ is clearly equivalent to the statement that neither of $D {-}D'$ or $D' {-}D$ induce an edge of ${{A^{\rm c}}}$. That $D {-}D'$ induces no edge in ${{A^{\rm c}}}$ is equivalent to the statement that for all $(y, x)$ in ${{A^{\rm c}}}$ with $x,y$ in $D$, either $y \not\in D'$ or $x \not\in D'$. As $D$ and $D'$ 