The clone of operations preserving a cycle with loops

We consider all the binary relations on a k-element set which, when viewed as directed graphs, consist of a k-cycle and some loops. If k ≥ 3 and the relation has at least two loops, we show that it is only preserved by essentially unary operations. In all other cases, the relation is preserved by operations that depend on a greater number of variables. Received March 9, 2006; accepted in final form September 14, 2007.