  Let $S\_{N}(P)$ be the poset obtained by adding a dummy vertex on each
diagonal edge of the $N$'s of a finite poset $P$. We show that
$S\_{N}(S\_{N}(P))$ is $N$-free. It follows that this poset is the smallest
$N$-free barycentric subdivision of the diagram of $P$, poset whose existence
was proved by P.A. Grillet. This is also the poset obtained by the algorithm
starting with $P\_0:=P$ and consisting at step $m$ of adding a dummy vertex on
a diagonal edge of some $N$ in $P\_m$, proving that the result of this
algorithm does not depend upon the particular choice of the diagonal edge
choosen at each step. These results are linked to drawing of posets.
