The Sperner property for polygonal graphs

A finite partially ordered set (poset) is said to have the Sperner property if at least one of its maximum antichains is formed from elements of the same hight. A polygonal graph is a directed acyclic graph derived from a circuit by some orientation of its edges. The reachability relation of a polygonal graph is a partial order. A criterion is presented for posets associated with polygonal graphs to have the Sperner property.