  Call a finite relational structure $k$-Slupecki if its only surjective
$k$-ary polymorphisms are essentially unary, and Slupecki if it is $k$-Slupecki
for all $k \geq 2$. We present conditions, some necessary and some sufficient,
for a reflexive digraph to be Slupecki. We prove that all digraphs that
triangulate a 1-sphere are Slupecki, as are all the ordinal sums $m \oplus n$
($m,n \geq 2$). We prove that the posets $P = m \oplus n \oplus k$ are not
3-Slupecki for $m,n,k \geq 2$, and prove there is a bound $B(m,k)$ such that
$P$ is 2-Slupecki if and only if $n > B(m,k)+1$; in particular there exist
posets that are 2-Slupecki but not 3-Slupecki.
