Let H be a hexagonal system.The z-transformation graph Z(H)of H is thegraph where the vertices are the perfect matchings of H and where two perfect mat-chings are joined by an edge provided their symmetric difference is a hexagon of H.In(2)it was proved that the connectivity x(Z(H))ofZ(H)is equal to δ(Z(H)).Furthermore

in this paper we determine a class H of hexagonal systems withk(Z(H))=I and subclass H2 of H such that every hexagonal system in H2 hasexactly two vertices of degree one.Finally the enumeration problem of H2 is con-sidered.